ContentsArticles

Invariant theory 1Bracket algebra 4Capelli's identity 5Catalecticant 13Cayley's Ω process 14Chevalley–Shephard–Todd theorem 15The Classical Groups 17Differential invariant 18Geometric invariant theory 20Gröbner basis 23Haboush's theorem 27Hall algebra 29Hilbert's basis theorem 30Hilbert's fourteenth problem 32Hilbert's syzygy theorem 33Hodge bundle 34Invariant estimator 35Invariant of a binary form 38Invariant polynomial 41Invariants of tensors 41Kostant polynomial 43Littlewood–Richardson rule 47Modular invariant of a group 54Moduli space 55Molien series 59Newton's identities 60Polynomial ring 68Quantum invariant 75Radical polynomial 76Reynolds operator 77Riemann invariant 79Ring of symmetric functions 81Schur polynomial 87Symbolic method 90

Trace identity 92Transvectant 92

ReferencesArticle Sources and Contributors 93Image Sources, Licenses and Contributors 95

Article LicensesLicense 96

Invariant theory 1

Invariant theoryInvariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties from the pointof view of their effect on functions. Classically, the theory dealt with the question of explicit description ofpolynomial functions that do not change, or are invariant, under the transformations from a given linear group.Invariant theory of finite groups has intimate connections with Galois theory. One of the first major results was themain theorem on the symmetric functions that described the invariants of the symmetric group Sn acting on thepolynomial ring R[x1, …, xn] by permutations of the variables. More generally, the Chevalley–Shephard–Toddtheorem characterizes finite groups whose algebra of invariants is a polynomial ring. Modern research in invarianttheory of finite groups emphasizes "effective" results, such as explicit bounds on the degrees of the generators. Thecase of positive characteristic, ideologically close to modular representation theory, is an area of active study, withlinks to algebraic topology.Invariant theory of infinite groups is inextricably linked with the development of linear algebra, especially, thetheories of quadratic forms and determinants. Another subject with strong mutual influence was projective geometry,where invariant theory was expected to play a major role in organizing the material. One of the highlights of thisrelationship is the symbolic method. Representation theory of semisimple Lie groups has its roots in invariant theory.David Hilbert's work on the question of the finite generation of the algebra of invariants (1890) resulted in thecreation of a new mathematical discipline, abstract algebra. A later paper of Hilbert (1893) dealt with the samequestions in more constructive and geometric ways, but remained virtually unknown until David Mumford broughtthese ideas back to life in the 1960s, in a considerably more general and modern form, in his geometric invarianttheory. In large measure due to the influence of Mumford, the subject of invariant theory is presently seen toencompass the theory of actions of linear algebraic groups on affine and projective varieties. A distinct strand ofinvariant theory, going back to the classical constructive and combinatorial methods of the nineteenth century, hasbeen developed by Gian-Carlo Rota and his school. A prominent example of this circle of ideas is given by thetheory of standard monomials.

The nineteenth century originsThe theory of invariants came into existence about the middle of the nineteenth century somewhat like Minerva: a grown-up virgin,mailed in the shining armor of algebra, she sprang forth from Cayley's Jovian head.

Weyl (1939b, p.489)

Classically, the term "invariant theory" refers to the study of invariant algebraic forms (equivalently, symmetrictensors) for the action of linear transformations. This was a major field of study in the latter part of the nineteenthcentury. Current theories relating to the symmetric group and symmetric functions, commutative algebra, modulispaces and the representations of Lie groups are rooted in this area.In greater detail, given a finite-dimensional vector space V of dimension n we can consider the symmetric algebraS(Sr(V)) of the polynomials of degree r over V, and the action on it of GL(V). It is actually more accurate to considerthe relative invariants of GL(V), or representations of SL(V), if we are going to speak of invariants: that is because ascalar multiple of the identity will act on a tensor of rank r in S(V) through the r-th power 'weight' of the scalar. Thepoint is then to define the subalgebra of invariants I(Sr(V)) for the action. We are, in classical language, looking atinvariants of n-ary r-ics, where n is the dimension of V. (This is not the same as finding invariants of GL(V) onS(V); this is an uninteresting problem as the only such invariants are constants.) The case that was most studied wasinvariants of binary forms where n=2.Other work included that of Felix Klein in computing the invariant rings of finite group actions on (the binarypolyhedral groups, classified by the ADE classification); these are the coordinate rings of du Val singularities.

Invariant theory 2

Like the Arabian phoenix rising out of its ashes, the theory of invariants, pronounced dead at the turn of the century, is once again atthe forefront of mathematics.

Kung & Rota (1984, p.27)

The work of David Hilbert, proving that I(V) was finitely presented in many cases, almost put an end to classicalinvariant theory for several decades, though the classical epoch in the subject continued to the final publications ofAlfred Young, more than 50 years later. Explicit calculations for particular purposes have been known in moderntimes (for example Shioda, with the binary octavics).

Hilbert's theoremsHilbert (1890) proved that if V is a finite dimensional representation of the complex algebraic group G= SLn(C) thenthe ring of invariants of G acting on the ring of polynomials R = S(V) is finitely generated. His proof used theReynolds operator ρ from R to RG with the properties• ρ(1)=1• ρ(a+b) = ρ(a)+ρ(b)• ρ(ab) = a ρ(b) whenever a is an invariant.Hilbert constructed the Reynolds operator explicitly using Cayley's omega process Ω, though now it is morecommon to construct ρ indirectly as follows: for compact groups G, the Reynolds operator is given by taking theaverage over G, and non-compact reductive groups can be reduced to the case of compact groups using Weyl'sunitarian trick.Given the Reynolds operator, Hilbert's theorem is proved as follows. The ring R is a polynomial ring so is graded bydegrees, and the ideal I is defined to be the ideal generated by the homogeneous invariants of positive degrees. ByHilbert's basis theorem the ideal I is finitely generated (as an ideal). Hence, I is finitely generated by finitely manyinvariants of G (because if we are given any - possibly infinite - subset S that generates a finitely generated ideal I,then I is already generated by some finite subset of S). Let i1,...,in be a finite set of invariants of G generating I (as anideal). The key idea is to show that these generate the ring RG of invariants. Suppose that x is some homogeneousinvariant of degree d>0. Then

x = a1i1 + ... + aninfor some aj in the ring R because x is in the ideal I. We can assume that aj is homogeneous of degree forevery j (otherwise, we replace aj by its homogeneous component of degree ; if we do this for every j, theequation x = a1i1 + ... + anin will remain valid). Now, applying the Reynolds operator to x = a1i1 + ... + anin gives

x = ρ(a1)i1 + ... + ρ(an)inWe are now going to show that x lies in the R-algebra generated by i1,...,in.First, let us do this in the case when the elements ρ(ak) all have degree less than d. In this case, they are all in theR-algebra generated by i1,...,in (by our induction assumption). Therefore x is also in this R-algebra (since x = ρ(a1)i1+ ... + ρ(an)in).In the general case, we cannot be sure that the elements ρ(ak) all have degree less than d. But we can replace eachρ(ak) by its homogeneous component of degree . As a result, these modified ρ(ak) are still G-invariants(because every homogeneous component of a G-invariant is a G-invariant) and have degree less than d (since

). The equation x = ρ(a1)i1 + ... + ρ(an)in still holds for our modified ρ(ak), so we can again concludethat x lies in the R-algebra generated by i1,...,in.Hence, by induction on the degree, all elements of RG are in the R-algebra generated by i1,...,in.

Invariant theory 3

Geometric invariant theoryThe modern formulation of geometric invariant theory is due to David Mumford, and emphasizes the construction ofa quotient by the group action that should capture invariant information through its coordinate ring. It is a subtletheory, in that success is obtained by excluding some 'bad' orbits and identifying others with 'good' orbits. In aseparate development the symbolic method of invariant theory, an apparently heuristic combinatorial notation, hasbeen rehabilitated.

References• Dieudonné, Jean A.; Carrell, James B. (1970), "Invariant theory, old and new", Advances in Mathematics 4: 1–80,

doi:10.1016/0001-8708(70)90015-0, ISSN 0001-8708, MR0255525 Reprinted as Dieudonné, Jean A.; Carrell,James B. (1971), "Invariant theory, old and new", Advances in Mathematics (Boston, MA: Academic Press) 4:1–80, doi:10.1016/0001-8708(70)90015-0, ISBN 978-0-12-215540-6, MR0279102

• Dolgachev, Igor (2003), Lectures on invariant theory, London Mathematical Society Lecture Note Series, 296,Cambridge University Press, doi:10.1017/CBO9780511615436, ISBN 978-0-521-52548-0, MR2004511

• Grace, J. H.; and Young, Alfred (1903), The algebra of invariants, Cambridge: Cambridge University Press Aclassic monograph.

• Grosshans, Frank D. (1997), Algebraic homogeneous spaces and invariant theory, New York: Springer,ISBN 3-540-63628-5

• Kung, Joseph P. S.; Rota, Gian-Carlo (1984), "The invariant theory of binary forms" [1], American MathematicalSociety. Bulletin. New Series 10 (1): 27–85, doi:10.1090/S0273-0979-1984-15188-7, ISSN 0002-9904,MR722856

• Hilbert, David (1890), "Ueber die Theorie der algebraischen Formen", Mathematische Annalen 36 (4): 473–534,doi:10.1007/BF01208503, ISSN 0025-5831

• Hilbert, D. (1893), "Über die vollen Invariantensysteme (On Full Invariant Systems)", Math. Annalen 42 (3): 313,doi:10.1007/BF01444162

• Neusel, Mara D.; and Smith, Larry (2002), Invariant Theory of Finite Groups, Providence, RI: AmericanMathematical Society, ISBN 0-8218-2916-5 A recent resource for learning about modular invariants of finitegroups.

• Olver, Peter J. (1999), Classical invariant theory, Cambridge: Cambridge University Press, ISBN 0-521-55821-2An undergraduate level introduction to the classical theory of invariants of binary forms, including the Omegaprocess starting at page 87.

• Popov, V.L. (2001), "Invariants, theory of" [2], in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,ISBN 978-1556080104

• Springer, T. A. (1977), Invariant Theory, New York: Springer, ISBN 0-387-08242-5 An older but still usefulsurvey.

• Sturmfels, Bernd (1993), Algorithms in Invariant Theory, New York: Springer, ISBN 0-387-82445-6 A beautifulintroduction to the theory of invariants of finite groups and techniques for computing them using Gröbner bases.

• Weyl, Hermann (1939), [[The Classical Groups|The Classical Groups. Their Invariants and Representations[3]]], Princeton University Press, ISBN 978-0-691-05756-9, MR0000255

• Weyl, Hermann (1939b), "Invariants" [4], Duke Mathematical Journal 5 (3): 489–502,doi:10.1215/S0012-7094-39-00540-5, ISSN 0012-7094, MR0000030

Invariant theory 4

External links• H. Kraft, C. Procesi, Classical Invariant Theory, a Primer [5]

References[1] http:/ / www. ams. org/ journals/ bull/ 1984-10-01/ S0273-0979-1984-15188-7[2] http:/ / www. encyclopediaofmath. org/ index. php?title=i/ i052350[3] http:/ / books. google. com/ books?isbn=0691057567[4] http:/ / projecteuclid. org/ euclid. dmj/ 1077491405[5] http:/ / www. math. unibas. ch/ ~kraft/ Papers/ KP-Primer. pdf

Bracket algebraA bracket algebra is an algebraic system that connects the notion of a supersymmetry algebra with a symbolicrepresentation of projective invariants.Given that L is a proper signed alphabet and Super[L] is the supersymmetric algebra, the bracket algebra Bracket[L]of dimension n over the field K is the quotient of the algebra Brace{L} obtained by imposing the congruencerelations below, where w, w', ..., w" are any monomials in Super[L]:1. {w} = 0 if length(w) ≠ n2. {w}{w'}...{w"} = 0 whenever any positive letter a of L occurs more than n times in the monomial

{w}{w'}...{w"}.3. Let {w}{w'}...{w"} be a monomial in Brace{L} in which some positive letter a occurs more than n times, and let

b, c, d, e, ..., f, g be any letters in L.

References• Anick, David; Rota, Gian-Carlo (September 15, 1991), "Higher-Order Syzygies for the Bracket Algebra and for

the Ring of Coordinates of the Grassmanian", Proceedings of the National Academy of Sciences 88 (18):8087–8090, doi:10.1073/pnas.88.18.8087, ISSN 0027-8424, JSTOR 2357546.

• Huang, Rosa Q.; Rota, Gian-Carlo; Stein, Joel A. (1990), "Supersymmetric Bracket Algebra and InvariantTheory" [1], Acta Applicandae Mathematicae (Kluwer Academic Publishers) 21: 193–246,doi:10.1007/BF00053298.

References[1] http:/ / www. springerlink. com/ content/ q821633w3291351g/

Capelli's identity 5

Capelli's identityIn mathematics, Capelli's identity, named after Alfredo Capelli (1887), is an analogue of the formuladet(AB) = det(A) det(B), for certain matrices with noncommuting entries, related to the representation theory of theLie algebra . It can be used to relate an invariant ƒ to the invariant Ωƒ, where Ω is Cayley's Ω process.

StatementSuppose that xij for i,j = 1,...,n are commuting variables. Write Eij for the polarization operator

The Capelli identity states that the following differential operators, expressed as determinants, are equal:

Both sides are differential operators. The determinant on the left has non-commuting entries, and is expanded withall terms preserving their "left to right" order. Such a determinant is often called a column-determinant, since it canbe obtained by the column expansion of the determinant starting from the first column. It can be formally written as

where in the product first come the elements from the first column, then from the second and so on. The determinanton the far right is Cayley's omega process, and the one on the left is the Capelli determinant.The operators Eij can be written in a matrix form:

where are matrices with elements Eij, xij, respectively. If all elements in these matrices would be

commutative then clearly . The Capelli identity shows that despitenoncommutativity there exists a "quantization" of the formula above. The only price for the noncommutivity is asmall correction: on the left hand side. For generic noncommutative matrices formulas like

do not exist, and the notion of the 'determinant' itself does not make sense for generic noncommutative matrices.That is why the Capelli identity still holds some mystery, despite many proofs offered for it. A very short proof doesnot seem to exist. Direct verification of the statement can be given as an exercise for n' = 2, but is already long for n= 3.

Capelli's identity 6

Relations with representation theoryConsider the following slightly more general context. Suppose that n and m are two integers and xij for i = 1,...,n,j =1,...,m, be commuting variables. Redefine Eij by almost the same formula:

with the only difference that summation index a ranges from 1 to m. One can easily see that such operators satisfythe commutation relations:

Here denotes the commutator . These are the same commutation relations which are satisfied by thematrices which have zeros everywhere except the position (i,j), where 1 stands. ( are sometimes calledmatrix units). Hence we conclude that the correspondence defines a representation of the Liealgebra in the vector space of polynomials of xij.

Case m = 1 and representation Sk Cn

It is especially instructive to consider the special case m = 1; in this case we have xi1, which is abbreviated as xi:

In particular, for the polynomials of the first degree it is seen that:

Hence the action of restricted to the space of first-order polynomials is exactly the same as the action of matrix

units on vectors in . So, from the representation theory point of view, the subspace of polynomials of firstdegree is a subrepresentation of the Lie algebra , which we identified with the standard representation in .Going further, it is seen that the differential operators preserve the degree of the polynomials, and hence thepolynomials of each fixed degree form a subrepresentation of the Lie algebra . One can see further that thespace of homogeneous polynomials of degree k can be identified with the symmetric tensor power of thestandard representation .One can also easily identify the highest weight structure of these representations. The monomial is a highestweight vector, indeed: for i < j. Its highest weight equals to (k, 0, ... ,0), indeed: .

Such representation is sometimes called bosonic representation of . Similar formulas define the

so-called fermionic representation, here are anti-commuting variables. Again polynomials of k-th degree form anirreducible subrepresentation which is isomorphic to i.e. anti-symmetric tensor power of . Highestweight of such representation is (0, ..., 0, 1, 0, ..., 0). These representations for k = 1, ..., n are fundamentalrepresentations of .

Capelli identity for m = 1

Let us return to the Capelli identity. One can prove the following:

the motivation for this equality is the following: consider for some commuting variables . Thematrix is of rank one and hence its determinant is equal to zero. Elements of matrix are defined by thesimilar formulas, however, its elements do not commute. The Capelli identity shows that the commutative identity:

can be preserved for the small price of correcting matrix by .Let us also mention that similar identity can be given for the characteristic polynomial:

Capelli's identity 7

where . The commutative counterpart of this is a simple fact that for rank = 1matrices the characteristic polynomial contains only the first and the second coefficients.Let us consider an example for n = 2.

Using

we see that this is equal to:

The universal enveloping algebra and its center

An interesting property of the Capelli determinant is that it commutes with all operators Eij, that is the commutatoris equal to zero. It can be generalized:

Consider any elements Eij in any ring, such that they satisfy the commutation relation, (so they can be differential operators above, matrix units eij or any other

elements) define elements Ck as follows:

where then:• elements Ck commute with all elements Eij• elements Ck can be given by the formulas similar to the commutative case:

i.e. they are sums of principal minors of the matrix E, modulo the Capelli correction . In particularelement C0 is the Capelli determinant considered above.These statements are interrelated with the Capelli identity, as will be discussed below, and similarly to it the directfew lines short proof does not seem to exist, despite the simplicity of the formulation.The universal enveloping algebra

can defined as an algebra generated byEij

subject to the relations

alone. The proposition above shows that elements Ckbelong to the center of . It can be shown that theyactually are free generators of the center of . They are sometimes called Capelli generators. The Capelliidentities for them will be discussed below.Consider an example for n = 2.

Capelli's identity 8

It is immediate to check that element commute with . (It corresponds to an obvious fact that theidentity matrix commute with all other matrices). More instructive is to check commutativity of the second elementwith . Let us do it for :

We see that the naive determinant will not commute with and the Capelli's correctionis essential to ensure the centrality.

General m and dual pairsLet us return to the general case:

for arbitrary n and m. Definition of operators Eij can be written in a matrix form: , where is

matrix with elements ; is matrix with elements ; is matrix with elements .

Capelli–Cauchy–Binet identities

For general m matrix E is given as product of the two rectangular matrices: X and transpose to D. If all elements ofthese matrices would commute then one knows that the determinant of E can be expressed by the so-calledCauchy–Binet formula via minors of X and D. An analogue of this formula also exists for matrix E again for thesame mild price of the correction :

,

In particular (similar to the commutative case): if m<n, then ; if m=n we return to theidentity above.Let us also mention that similar to the commutative case (see Cauchy–Binet for minors), one can express not onlythe determinant of E, but also its minors via minors of X and D:

,

Here K = (k1 < k2 < ... < ks), L = (l1 < l2 < ... < ls), are arbitrary multi-indexes; as usually denotes a submatrixof M formed by the elements M kalb. Pay attention that the Capelli correction now contains s, not n as in previousformula. Note that for s=1, the correction (s − i) disappears and we get just the definition of E as a product of X andtranspose to D. Let us also mention that for generic K,L corresponding minors do not commute with all elements Eij,so the Capelli identity exists not only for central elements.As a corollary of this formula and the one for the characteristic polynomial in the previous section let us mention thefollowing:

where . This formula is similar to thecommutative case, modula at the left hand side and t[n] instead of tn at the right hand side.

Capelli's identity 9

Relation to dual pairs

Modern interest in these identities has been much stimulated by Roger Howe who considered them in his theory ofreductive dual pairs (also known as Howe duality). To make the first contact with these ideas, let us look moreprecisely on operators . Such operators preserve the degree of polynomials. Let us look at the polynomials ofdegree 1: , we see that index l is preserved. One can see that from the representation theory pointof view polynomials of the first degree can be identified with direct sum of the representations ,here l-th subspace (l=1...m) is spanned by , i = 1, ..., n. Let us give another look on this vector space:

Such point of view gives the first hint of symmetry between m and n. To deepen this idea let us consider:

These operators are given by the same formulas as modula renumeration , hence by the same argumentswe can deduce that form a representation of the Lie algebra in the vector space of polynomials of xij.

Before going further we can mention the following property: differential operators commute with differential

operators .The Lie group acts on the vector space in a natural way. One can show that thecorresponding action of Lie algebra is given by the differential operators and respectively.

This explains the commutativity of these operators.The following deeper properties actually hold true:

• The only differential operators which commute with are polynomials in , and vice versa.• Decomposition of the vector space of polynomials into a direct sum of tensor products of irreducible

representations of and can be given as follows:

The summands are indexed by the Young diagrams D, and representations are mutually non-isomorphic. Anddiagram determine and vice versa.• In particular the representation of the big group is multiplicity free, that is each irreducible

representation occurs only one time.One easily observe the strong similarity to Schur–Weyl duality.

GeneralizationsMuch work have been done on the identity and its generalizations. Approximately two dozens of mathematicians andphysicists contributed to the subject, to name a few: R. Howe, B. Kostant[1] [2] Fields medalist A. Okounkov[3] [4] A.Sokal,[5] D. Zeilberger.[6]

It seems historically the first generalizations were obtained by Herbert Westren Turnbull in 1948,[7] who found thegeneralization for the case of symmetric matrices (see[5] [6] for modern treatments).The other generalizations can be divided into several patterns. Most of them are based on the Lie algebra point of view. Such generalizations consist of changing Lie algebra to simple Lie algebras [8] and their super[9] [10] (q),[11] [12] and current versions.[13] As well as identity can be generalized for different reductive dual pairs.[14] [15]

And finally one can consider not only the determinant of the matrix E, but its permanent,[16] trace of its powers and immanants.[3] [4] [17] [18] Let us mention few more papers;[19] [20] [21] still the list of references is incomplete. It has been believed for quite a long time that the identity is intimately related with semi-simple Lie algebras. Surprisingly a new purely algebraic generalization of the identity have been found in 2008[5] by S. Caracciolo, A. Sportiello, A.

Capelli's identity 10

D. Sokal which has nothing to do with any Lie algebras.

Turnbull's identity for symmetric matricesConsider symmetric matrices

Herbert Westren Turnbull[7] in 1948 discovered the following identity:

Combinatorial proof can be found in the paper,[6] another proof and amusing generalizations in the paper,[5] see alsodiscussion below.

The Howe–Umeda–Kostant–Sahi identity for antisymmetric matricesConsider antisymmetric matrices

Then

The Caracciolo–Sportiello–Sokal identity for Manin matricesConsider two matrices M and Y over some associative ring which satisfy the following condition

for some elements Qil. Or ”in words”: elements in j-th column of M commute with elements in k-th row of Y unlessj = k, and in this case commutator of the elements Mik and Ykl depends only on i, l, but does not depend on k.Assume that M is a Manin matrix (the simplest example is the matrix with commuting elements).Then for the square matrix case

Here Q is a matrix with elements Qil, and diag(n − 1, n − 2, ..., 1, 0) means the diagonal matrix with the elementsn − 1, n − 2, ..., 1, 0 on the diagonal.See [5] proposition 1.2' formula (1.15) page 4, our Y is transpose to their B.Obviously the original Cappeli's identity the particular case of this identity. Moreover from this identity one can seethat in the original Capelli's identity one can consider elements

for arbitrary functions fij and the identity still will be true.

Capelli's identity 11

The Mukhin–Tarasov–Varchenko identity and the Gaudin model

Statement

Consider matrices X and D as in Capelli's identity, i.e. with elements and at position (ij).Let z be another formal variable (commuting with x). Let A and B be some matrices which elements are complexnumbers.

Here the first determinant is understood (as always) as column-determinant of a matrix with non-commutativeentries. The determinant on the right is calculated as if all the elements commute, and putting all x and z on the left,while derivations on the right. (Such recipe is called a Wick ordering in the quantum mechanics).

The Gaudin quantum integrable system and Talalaev's theorem

The matrix

is a Lax matrix for the Gaudin quantum integrable spin chain system. D. Talalaev solved the long-standing problemof the explicit solution for the full set of the quantum commuting conservation laws for the Gaudin model,discovering the following theorem.Consider

Then for all i,j,z,w

i.e. Hi(z) are generating functions in z for the differential operators in x which all commute. So they provide quantumcommuting conservation laws for the Gaudin model.

Permanents, immanants, traces – "higher Capelli identities"The original Capelli identity is a statement about determinants. Later, analogous identities were found forpermanents, immanants and traces.

Turnbull's identity for permanents of antisymmetric matrices

Consider the antisymmetric matrices X and D with elements xij and corresponding derivations, as in the case of theHUKS identity above.Then

Let us cite [6] : "...is stated without proof at the end of Turnbull’s paper". The authors themselves follow Turnbull –at the very end of their paper they write:"Since the proof of this last identity is very similar to the proof of Turnbull’s symmetric analog (with a slight twist),we leave it as an instructive and pleasant exercise for the reader.".

Capelli's identity 12

References

Inline[1] Kostant, B.; Sahi, S. (1991), "The Capelli Identity, tube domains, and the generalized Laplace transform", Advances in Math. 87: 71–92,

doi:10.1016/0001-8708(91)90062-C[2] Kostant, B.; Sahi, S. (1993), "Jordan algebras and Capelli identities", Inventiones Mathematicae 112 (1): 71–92, doi:10.1007/BF01232451[3] Okounkov, A. (1996), Quantum Immanants and Higher Capelli Identities, arXiv:q-alg/9602028[4] Okounkov, A. (1996), Young Basis, Wick Formula, and Higher Capelli Identities, arXiv:q-alg/9602027[5] Caracciolo, S.; Sportiello, A.; Sokal, A. (2008), Noncommutative determinants, Cauchy–Binet formulae, and Capelli-type identities. I.

Generalizations of the Capelli and Turnbull identities, arXiv:0809.3516[6] Foata, D.; Zeilberger, D. (1993), Combinatorial Proofs of Capelli's and Turnbull's Identities from Classical Invariant Theory,

arXiv:math/9309212[7] Turnbull, Herbert Westren (1948), "Symmetric determinants and the Cayley and Capelli operators", Proc. Edinburgh Math. Soc. 8 (2):

76–86, doi:10.1017/S0013091500024822[8] Molev, A.; Nazarov, M. (1997), Capelli Identities for Classical Lie Algebras, arXiv:q-alg/9712021[9] Molev, A. (1996), Factorial supersymmetric Schur functions and super Capelli identities, arXiv:q-alg/9606008[10] Nazarov, M. (1996), Capelli identities for Lie superalgebras, arXiv:q-alg/9610032[11] Noumi, M.; Umeda, T.; Wakayma, M. (1994), "A quantum analogue of the Capelli identity and an elementary differential calculus on

GLq(n)" (http:/ / projecteuclid. org/ euclid. dmj/ 1077286975), Duke Mathematical Journal 76 (2): 567–594,doi:10.1215/S0012-7094-94-07620-5,

[12] Noumi, M.; Umeda, T.; Wakayma, M. (1996), "Dual pairs, spherical harmonics and a Capelli identity in quantum group theory" (http:/ /www. numdam. org/ item?id=CM_1996__104_3_227_0), Compositio Mathematica 104 (2): 227–277,

[13] Mukhin, E.; Tarasov, V.; Varchenko, A. (2006), A generalization of the Capelli identity, arXiv:math.QA/0610799[14] Itoh, M. (2004), "Capelli identities for reductive dual pairs", Advances in Mathematics 194 (2): 345–397, doi:10.1016/j.aim.2004.06.010[15] Itoh, M. (2005), "Capelli Identities for the dual pair ( O M, Sp N)", Mathematische Zeitschrift 246 (1–2): 125–154,

doi:10.1007/s00209-003-0591-2[16] Nazarov, M. (1991), "Quantum Berezinian and the classical Capelli identity", Letters in Mathematical Physics 21 (2): 123–131,

doi:10.1007/BF00401646[17] Nazarov, M. (1996), Yangians and Capelli identities, arXiv:q-alg/9601027[18] Molev, A. (1996), A Remark on the Higher Capelli Identities, arXiv:q-alg/9603007[19] Kinoshita, K.; Wakayama, M. (2002), "Explicit Capelli identities for skew symmetric matrices", Proceedings of the Edinburgh

Mathematical Society 45 (2): 449–465, doi:10.1017/S0013091500001176[20] Hashimoto, T. (2008), Generating function for GLn-invariant differential operators in the skew Capelli identity, arXiv:0803.1339[21] Nishiyama, K.; Wachi, A. (2008), A note on the Capelli identities for symmetric pairs of Hermitian type, arXiv:0808.0607

General• Capelli, Alfredo (1887), "Ueber die Zurückführung der Cayley'schen Operation Ω auf gewöhnliche

Polar-Operationen", Mathematische Annalen (Berlin / Heidelberg: Springer) 29 (3): 331–338,doi:10.1007/BF01447728, ISSN 1432-1807

• Howe, Roger (1989), "Remarks on classical invariant theory", Transactions of the American MathematicalSociety (American Mathematical Society) 313 (2): 539–570, doi:10.2307/2001418, ISSN 0002-9947,JSTOR 2001418, MR0986027

• Howe, Roger; Umeda, Toru (1991), "The Capelli identity, the double commutant theorem, and multiplicity-freeactions", Mathematische Annalen 290 (1): 565–619, doi:10.1007/BF01459261

• Umeda, Tôru (1998), "The Capelli identities, a century after" (http:/ / books. google. com/books?isbn=0821808400), Selected papers on harmonic analysis, groups, and invariants, Amer. Math. Soc.Transl. Ser. 2, 183, Providence, R.I.: Amer. Math. Soc., pp. 51–78, ISBN 978-0-8218-0840-5, MR1615137

• Weyl, Hermann (1946), The Classical Groups: Their Invariants and Representations (http:/ / books. google. com/?id=zmzKSP2xTtYC), Princeton University Press, ISBN 978-0-691-05756-9, MR0000255, retrieved 03/2007/26

Catalecticant 13

CatalecticantBut the catalecticant of the biquadratic function of x, y was first brought into notice as an invariant by Mr Boole; and thediscriminant of the quadratic function of x, y is identical with its catalecticant, as also with its Hessian. Meicatalecticizant wouldmore completely express the meaning of that which, for the sake of brevity, I denominate the catalecticant.

Sylvester (1852), quoted by (Miller 2010)

In mathematical invariant theory, the catalecticant of a binary form of degree 2n is a polynomial in its coefficientsthat vanishes when the binary form is a sum of at most n powers Sturmfels (1993). It was introduced by Sylvester(1852); see (Miller 2010). The word catalectic refers to an incomplete line of verse, lacking a syllable at the end orending with an incomplete foot.The catalecticant can be given as the determinant of a catalecticant matrix (Eisenbud 1988), also called a Hankelmatrix, that is a square matrix with constant (positive sloping) skew-diagonals, such as

References• Eisenbud, David (1988), "Linear sections of determinantal varieties", American Journal of Mathematics 110 (3):

541–575, doi:10.2307/2374622, ISSN 0002-9327, MR944327• Elliott, Edwin Bailey (1913) [1895], An introduction to the algebra of quantics. [1] (2nd ed.), Oxford. Clarendon

Press, JFM 26.0135.01• Sturmfels, Bernd (1993), Algorithms in invariant theory, Texts and Monographs in Symbolic Computation,

Berlin, New York: Springer-Verlag, doi:10.1007/978-3-211-77417-5, ISBN 978-3-211-82445-0, MR1255980• Sylvester, J. J. (1852), "On the principles of the calculus of forms", Cambridge and Dublin Mathematical

Journal: 52–97

External links• Miller, Jeff (2010), Earliest Known Uses of Some of the Words of Mathematics (C) [2]

References[1] http:/ / books. google. com/ books?id=Az5tAAAAMAAJ[2] http:/ / jeff560. tripod. com/ c. html

Cayley's Ω process 14

Cayley's Ω processIn mathematics, Cayley's Ω process, introduced by Arthur Cayley (1846), is a relatively invariant differentialoperator on the general linear group, that is used to construct invariants of a group action.As a partial differential operator acting on functions of n2 variables xij, the omega operator is given by thedeterminant

ApplicationsCayley's Ω process appears in Capelli's identity, which Weyl (1946) used to find generators for the invariants ofvarious classical groups acting on natural polynomial algebras.Hilbert (1890) used Cayley's Ω process in his proof of finite generation of rings of invariants of the general lineargroup. His use of the Ω process gives an explicit formula for the Reynolds operator of the special linear group.Cayley's Ω process is used to define transvectants.

References• Cayley, Arthur (1846), "On linear transformations" [1], Cambridge and Dublin mathematical journal 1: 104–122

Reprinted in Cayley (1889), The collected mathematical papers, 1, Cambridge: Cambridge University press,pp. 95–112

• Hilbert, David (1890), "Ueber die Theorie der algebraischen Formen", Mathematische Annalen 36 (4): 473–534,doi:10.1007/BF01208503, ISSN 0025-5831

• Howe, Roger (1989), "Remarks on classical invariant theory.", Transactions of the American MathematicalSociety (American Mathematical Society) 313 (2): 539–570, doi:10.1090/S0002-9947-1989-0986027-X,ISSN 0002-9947, JSTOR 2001418, MR0986027

• Olver, Peter J. (1999), Classical invariant theory, Cambridge University Press, ISBN 978-0-521-55821-1• Sturmfels, Bernd (1993), Algorithms in invariant theory, Texts and Monographs in Symbolic Computation,

Berlin, New York: Springer-Verlag, ISBN 978-3-211-82445-0, MR1255980• Weyl, Hermann (1946), The Classical Groups: Their Invariants and Representations [2], Princeton University

Press, ISBN 978-0-691-05756-9, MR0000255, retrieved 03/2007/26

References[1] http:/ / books. google. com/ books?id=PBcAAAAAMAAJ& dq=Cambridge%20and%20Dublin%20mathematical%20journal%201846&

pg=PR3#v=onepage& q=Cambridge%20and%20Dublin%20mathematical%20journal%201846& f=false[2] http:/ / books. google. com/ ?id=zmzKSP2xTtYC

ChevalleyShephardTodd theorem 15

Chevalley–Shephard–Todd theoremIn mathematics, the Chevalley–Shephard–Todd theorem in invariant theory of finite groups states that the ring ofinvariants of a finite group acting on a complex vector space is a polynomial ring if and only if the group isgenerated by pseudoreflections. In the case of subgroups of the complex general linear group the theorem was firstproved by G. C. Shephard and J. A. Todd (1954) who gave a case-by-case proof. Claude Chevalley (1955) soonafterwards gave a uniform proof. It has been extended to finite linear groups over an arbitrary field in thenon-modular case by Jean-Pierre Serre.

Statement of the theoremLet V be a finite-dimensional vector space over a field K and let G be a finite subgroup of the general linear groupGL(V). An element s of GL(V) is called a pseudoreflection if it fixes a codimension one subspace of V and is not theidentity transformation I, or equivalently, if the kernel Ker (s − I) has codimension one in V. Assume that the orderof G is relatively prime to the characteristic of K (the so-called non-modular case). Then the following threeproperties are equivalent:• The group G is generated by pseudoreflections.• The algebra of invariants K[V]G is a (free) polynomial algebra.• The algebra K[V] is a free module over K[V]G.In the case when the field K is the field C of complex numbers, the first condition is usually stated as "G is acomplex reflection group". Shephard and Todd derived a full classification of such groups.

Examples• Let V be one-dimensional. Then any finite group faithfully acting on V is a subgroup of the multiplicative group

of the field K, and hence a cyclic group. It follows that G consists of roots of unity of order dividing n, where n isits order, so G is generated by pseudoreflections. In this case, K[V] = K[x] is the polynomial ring in one variableand the algebra of invariants of G is the subalgebra generated by xn, hence it is a polynomial algebra.

• Let V = Kn be the standard n-dimensional vector space and G be the symmetric group Sn acting by permutations ofthe elements of the standard basis. The symmetric group is generated by transpositions (ij), which act byreflections on V. On the other hand, by the main theorem of symmetric functions, the algebra of invariants is thepolynomial algebra generated by the elementary symmetric functions e1, … en.

• Let V = K2 and G be the cyclic group of order 2 acting by ±I. In this case, G is not generated by pseudoreflections,since the nonidentity element s of G acts without fixed points, so that dim Ker (s − I) = 0. On the other hand, thealgebra of invariants is the subalgebra of K[V] = K[x, y] generated by the homogeneous elements x2, xy, and y2 ofdegree 2. This subalgebra is not a polynomial algebra because of the relation x2y2 = (xy)2.

ChevalleyShephardTodd theorem 16

GeneralizationsBroer (2007) gave an extension of the Chevalley–Shephard–Todd theorem to positive characteristic.There has been much work on the question of when a reductive algebraic group acting on a vector space has apolynomial ring of invariants. In the case when the algebraic group is simple and the representation is irreducible allcases when the invariant ring is polynomial have been classified by Schwarz (1978)In general, the ring of invariants of a finite group acting linearly on a complex vector space is Cohen-Macaulay, so itis a finite rank free module over a polynomial subring.

References• Broer, Abraham (2007), On Chevalley-Shephard-Todd's theorem in positive characteristic, [], arXiv:0709.0715• Chevalley, Claude (1955), "Invariants of finite groups generated by reflections", Amer. J. Of Math. 77 (4):

778–782, doi:10.2307/2372597, JSTOR 2372597• Neusel, Mara D.; Smith, Larry (2002), Invariant Theory of Finite Groups, American Mathematical Society,

ISBN 0-8218-2916-5• Shephard, G. C.; Todd, J. A. (1954), "Finite unitary reflection groups", Canadian J. Math. 6: 274–304,

doi:10.4153/CJM-1954-028-3• Schwarz, G. (1978), "Representations of simple Lie groups with regular rings of invariants", Invent. Math. 49 (2):

167–191, doi:10.1007/BF01403085• Smith, Larry (1997), "Polynomial invariants of finite groups. A survey of recent developments" [1], Bull. Amer.

Math. Soc. 34 (3): 211–250, doi:10.1090/S0273-0979-97-00724-6, MR1433171• Springer, T. A. (1977), Invariant Theory, Springer, ISBN 0-387-08242-5

References[1] http:/ / www. ams. org/ bull/ 1997-34-03/ S0273-0979-97-00724-6/

The Classical Groups 17

The Classical GroupsIn Weyl's wonderful and terrible1 book The Classical Groups [W] one may discern two main themes: first, the study of thepolynomial invariants for an arbitrary number of (contravariant or covariant) variables for a standard classical group action; second,the isotypic decomposition of the full tensor algebra for such an action.1Most people who know the book feel the material in it is wonderful. Many also feel the presentation is terrible. (The author is notamong these latter.)

Howe (1989, p.539)

In mathematics, The Classical Groups: Their Invariants and Representations is a book by Weyl (1939), whichdescribes classical invariant theory in terms of representation theory. It is largely responsible for the revival ofinterest in invariant theory, which had been almost killed off by Hilbert's solution of its main problems in the 1890s.Weyl (1939b) gave an informal talk about the topic of his book.

ContentsChapter I defines invariants and other basic ideas and describes the relation to Klein's Erlanger program in geometry.Chapter II describes the invariants of the special and general linear group of a vector space V on the polynomialsover a sum of copies of V and its dual. It uses the Capelli identity to find an explicit set of generators for theinvariants.Chapter III studies the group ring of a finite group and its decomposition into a sum of matrix algebras.Chapter 4 discusses Schur-Weyl duality between representations of the symmetric and general linear groups.Chapters V and VI extend the discussion of invariants of the general linear group in chapter II to the orthogonal andsymplectic groups, showing that the ring of invariants is generated by the obvious ones.Chapter VII describes the Weyl character formula for the characters of representations of the classical groups.Chapter VIII on invariant theory proves Hilbert's theorem that invariants of the special linear group are finitelygenerated.Chapter IX and X give some supplements to the previous chapters.

References• Howe, Roger (1988), "The classical groups and invariants of binary forms", in Wells, R. O. Jr., The mathematical

heritage of Hermann Weyl (Durham, NC, 1987), Proc. Sympos. Pure Math., 48, Providence, R.I.: AmericanMathematical Society, pp. 133–166, ISBN 978-0-8218-1482-6, MR974333

• Howe, Roger (1989), "Remarks on classical invariant theory.", Transactions of the American MathematicalSociety (American Mathematical Society) 313 (2): 539–570, doi:10.2307/2001418, ISSN 0002-9947,JSTOR 2001418, MR0986027

• Jacobson, Nathan (1940), "Book Review: The Classical Groups" [1], Bulletin of the American MathematicalSociety 46 (7): 592–595, doi:10.1090/S0002-9904-1940-07236-2, ISSN 0002-9904, MR1564136

• Weyl, Hermann (1939), The Classical Groups. Their Invariants and Representations [3], Princeton UniversityPress, ISBN 978-0-691-05756-9, MR0000255

• Weyl, Hermann (1939), "Invariants" [4], Duke Mathematical Journal 5: 489–502, ISSN 0012-7094, MR0000030

The Classical Groups 18

References[1] http:/ / projecteuclid. org/ euclid. bams/ 1183502778

Differential invariantIn mathematics, a differential invariant is an invariant for the action of a Lie group on a space that involves thederivatives of graphs of functions in the space. Differential invariants are fundamental in projective differentialgeometry, and the curvature is often studied from this point of view.[1] Differential invariants were introduced inspecial cases by Sophus Lie in the early 1880s and studied by Georges Henri Halphen at the same time. Lie (1884)was the first general work on differential invariants, and established the relationship between differential invariants,invariant differential equations, and invariant differential operators.Differential invariants are contrasted with geometric invariants. Whereas differential invariants can involve adistinguished choice of independent variables (or a parameterization), geometric invariants do not. Élie Cartan'smethod of moving frames is a refinement that, while less general than Lie's methods of differential invariants, alwaysyields invariants of the geometrical kind.

DefinitionThe simplest case is for differential invariants for one independent variable x and one dependent variable y. Let G bea Lie group acting on R2. Then G also acts, locally, on the space of all graphs of the form y = ƒ(x). Roughly speaking,a k-th order differential invariant is a function

depending on y and its first k derivatives with respect to x, that is invariant under the action of the group.The group can act on the higher-order derivatives in a nontrivial manner that requires computing the prolongation ofthe group action. The action of G on the first derivative, for instance, is such that the chain rule continues to hold: if

then

Similar considerations apply for the computation of higher prolongations. This method of computing theprolongation is impractical, however, and it is much simpler to work infinitesimally at the level of Lie algebras andthe Lie derivative along the G action.More generally, differential invariants can be considered for mappings from any smooth manifold X into anothersmooth manifold Y for a Lie group acting on the Cartesian product X×Y. The graph of a mapping X → Y is asubmanifold of X×Y that is everywhere transverse to the fibers over X. The group G acts, locally, on the space ofsuch graphs, and induces an action on the k-th prolongation Y(k) consisting of graphs passing through each pointmodulo the relation of k-th order contact. A differential invariant is a function on Y(k) that is invariant under theprolongation of the group action.

Differential invariant 19

Applications• Differential invariants can be applied to the study of systems of partial differential equations: seeking similarity

solutions that are invariant under the action of a particular group can reduce the dimension of problem (a "reducedsystem").[2]

• Noether's theorem implies the existence of differential invariants corresponding to every differentiable symmetryof a variational problem.

• Flow characteristics using computer vision[3]

• Geometric integration

Notes[1] Guggenheimer 1977[2] Olver 1994, Chapter 3[3] http:/ / dspace. mit. edu/ bitstream/ handle/ 1721. 1/ 3348/ P-2219-29812804. pdf?sequence=1

References• Guggenheimer, Heinrich (1977), Differential Geometry, New York: Dover Publications,

ISBN 978-0-486-63433-3.• Lie, Sophus (1884), "Über Differentialinvarianten", Gesammelte Adhandlungen, 6, Leipzig: B.G. Teubner,

pp. 95–138; English translation: Ackerman, M; R (1975), Sophus Lie's 1884 Differential Invariant Paper,Brookline, Mass.: Math Sci Press.

• Olver, Peter J. (1993), Applications of Lie groups to differential equations (2nd ed.), Berlin, New York:Springer-Verlag, ISBN 978-0-387-94007-6.

• Olver, Peter J. (1995), Equivalents, Invariants, and Symmetry, Cambridge University Press,ISBN 978-0-521-47811-3.

• Mansfield, Elizabeth Louise (2009), A Practical Guide to the Invariant Calculus (http:/ / www. kent. ac. uk/ ims/personal/ elm2/ FrameBook2Jun09. pdf); to be published by Cambridge 2010, ISBN 9780521857017.

External links• Invariant Variation Problems (http:/ / www. physics. ucla. edu/ ~cwp/ articles/ noether. trans/ english/ mort186.

html)

Geometric invariant theory 20

Geometric invariant theoryIn mathematics Geometric invariant theory (or GIT) is a method for constructing quotients by group actions inalgebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas fromthe paper (Hilbert 1893) in classical invariant theory.Geometric invariant theory studies an action of a group G on an algebraic variety (or scheme) X and providestechniques for forming the 'quotient' of X by G as a scheme with reasonable properties. One motivation was toconstruct moduli spaces in algebraic geometry as quotients of schemes parametrizing marked objects. In the 1970sand 1980s the theory developed interactions with symplectic geometry and equivariant topology, and was used toconstruct moduli spaces of objects in differential geometry, such as instantons and monopoles.

BackgroundInvariant theory is concerned with a group action of a group G on an algebraic variety (or a scheme) X. Classicalinvariant theory addresses the situation when X = V is a vector space and G is either a finite group, or one of theclassical Lie groups that acts linearly on V. This action induces a linear action of G on the space of polynomialfunctions R(V) on V by the formula

The polynomial invariants of the G-action on V are those polynomial functions f on V which are fixed under the'change of variables' due to the action of the group, so that g·f = f for all g in G. They form a commutative algebraA = R(V)G, and this algebra is interpreted as the algebra of functions on the 'invariant theory quotient' V //G. In thelanguage of modern algebraic geometry,

Several difficulties emerge from this description. The first one, successfully tackled by Hilbert in the case of ageneral linear group, is to prove that the algebra A is finitely generated. This is necessary if one wanted the quotientto be an affine algebraic variety. Whether a similar fact holds for arbitrary groups G was the subject of Hilbert'sfourteenth problem, and Nagata demonstrated that the answer was negative in general. On the other hand, in thecourse of development of representation theory in the first half of the twentieth century, a large class of groups forwhich the answer is positive was identified; these are called reductive groups and include all finite groups and allclassical groups.The finite generation of the algebra A is but the first step towards the complete description of A, and the progress inresolving this more delicate question was rather modest. The invariants had classically been described only in arestricted range of situations, and the complexity of this description beyond the first few cases held out little hope forfull understanding of the algebras of invariants in general. Furthermore, it may happen that all polynomial invariantsf take the same value on a given pair of points u and v in V, yet these points are in different orbits of the G-action. Asimple example is provided by the multiplicative group C* of non-zero complex numbers that acts on ann-dimensional complex vector space Cn by scalar multiplication. In this case, every polynomial invariant is aconstant, but there are many different orbits of the action. The zero vector forms an orbit by itself, and the non-zeromultiples of any non-zero vector form an orbit, so that non-zero orbits are paramatrized by the points of the complexprojective space CPn−1. If this happens, one says that "invariants do not separate the orbits", and the algebra Areflects the topological quotient space X /G rather imperfectly. Indeed, the latter space is frequently non-separated. In1893 Hilbert formulated and proved a criterion for determining those orbits which are not separated from the zeroorbit by invariant polynomials. Rather remarkably, unlike his earlier work in invariant theory, which led to the rapiddevelopment of abstract algebra, this result of Hilbert remained little known and little used for the next 70 years.Much of the development of invariant theory in the first half of the twentieth century concerned explicitcomputations with invariants, and at any rate, followed the logic of algebra rather than geometry.

Geometric invariant theory 21

Mumford's bookGeometric invariant theory was founded and developed by Mumford in a monograph, first published in 1965, thatapplied ideas of nineteenth century invariant theory, including some results of Hilbert, to modern algebraic geometryquestions. (The book was greatly expanded in two later editions, with extra appendices by Fogarty and Mumford,and a chapter on symplectic quotients by Kirwan.) The book uses both scheme theory and computational techniquesavailable in examples. The abstract setting used is that of a group action on a scheme X. The simple-minded idea ofan orbit space

G\X,i.e. the quotient space of X by the group action, runs into difficulties in algebraic geometry, for reasons that areexplicable in abstract terms. There is in fact no general reason why equivalence relations should interact well withthe (rather rigid) regular functions (polynomial functions), such as are at the heart of algebraic geometry. Thefunctions on the orbit space G\X that should be considered are those on X that are invariant under the action of G.The direct approach can be made, by means of the function field of a variety (i.e. rational functions): take theG-invariant rational functions on it, as the function field of the quotient variety. Unfortunately this — the point ofview of birational geometry — can only give a first approximation to the answer. As Mumford put it in the Prefaceto the book:

The problem is, within the set of all models of the resulting birational class, there is one model whosegeometric points classify the set of orbits in some action, or the set of algebraic objects in some moduliproblem.

In Chapter 5 he isolates further the specific technical problem addressed, in a moduli problem of quite classical type— classify the big 'set' of all algebraic varieties subject only to being non-singular (and a requisite condition onpolarization). The moduli are supposed to describe the parameter space. For example for algebraic curves it has beenknown from the time of Riemann that there should be connected components of dimensions

0, 1, 3, 6, 9, …according to the genus g =0, 1, 2, 3, 4, … , and the moduli are functions on each component. In the coarse moduliproblem Mumford considers the obstructions to be:• non-separated topology on the moduli space (i.e. not enough parameters in good standing)• infinitely many irreducible components (which isn't avoidable, but local finiteness may hold)• failure of components to be representable as schemes, although respectable topologically.It is the third point that motivated the whole theory. As Mumford puts it, if the first two difficulties are resolved

[the third question] becomes essentially equivalent to the question of whether an orbit space of some locallyclosed subset of the Hilbert or Chow schemes by the projective group exists.

To deal with this he introduced a notion (in fact three) of stability. This enabled him to open up the previouslytreacherous area — much had been written, in particular by Francesco Severi, but the methods of the literature hadlimitations. The birational point of view can afford to be careless about subsets of codimension 1. To have a modulispace as a scheme is on one side a question about characterising schemes as representable functors (as theGrothendieck school would see it); but geometrically it is more like a compactification question, as the stabilitycriteria revealed. The restriction to non-singular varieties will not lead to a compact space in any sense as modulispace: varieties can degenerate to having singularities. On the other hand the points that would correspond to highlysingular varieties are definitely too 'bad' to include in the answer. The correct middle ground, of points stable enoughto be admitted, was isolated by Mumford's work. The concept was not entirely new, since certain aspects of it wereto be found in David Hilbert's final ideas on invariant theory, before he moved on to other fields.The book's Preface also enunciated the Mumford conjecture, later proved by William Haboush.

Geometric invariant theory 22

StabilityIf a reductive group G acts linearly on a vector space V, then a non-zero point of V is called• unstable if 0 is in the closure of its orbit,• semi-stable if 0 is not in the closure of its orbit,• stable if its orbit is closed, and its stabilizer is finite.There are equivalent ways to state these:• A non-zero point x is unstable if and only if there is a 1-parameter subgroup of G all of whose weights with

respect to x are positive.• A non-zero point x is unstable if and only if every invariant polynomial has the same value on 0 and x.• A non-zero point x is semistable if and only if there is no 1-parameter subgroup of G all of whose weights with

respect to x are positive.• A non-zero point x is semistable if and only if some invariant polynomial has different values on 0 and x.• A non-zero point x is stable if and only if every 1-parameter subgroup of G has positive (and negative) weights

with respect to x.• A non-zero point x is stable if and only if for every y not in the orbit of x there is some invariant polynomial that

has different values on y and x, and the ring of invariant polynomials has transcendence degree dim(V)−dim(G).A point of the corresponding projective space of V is called unstable, semi-stable, or stable if it is the image of apoint in V with the same property. "Unstable" is the opposite of "semistable" (not "stable"). The unstable points forma Zariski closed set of projective space, while the semistable and stable points both form Zariski open sets (possiblyempty). These definitions are from (Mumford 1977) and are not equivalent to the ones in the first edition ofMumford's book.Many moduli spaces can be constructed as the quotients of the space of stable points of some subset of projectivespace by some group action. These spaces can often by compactified by adding certain equivalence classes ofsemistable points. Different stable orbits correspond to different points in the quotient, but two different semistableorbits may correspond to the same point in the quotient if their closures intersect.Example: (Deligne & Mumford 1969) A stable curve is a reduced connected curve of genus ≥2 such that its onlysingularities are ordinary double points and every non-singular rational component meets the other components in atleast 3 points. The moduli space of stable curves of genus g is the quotient of a subset of the Hilbert scheme ofcurves in P5g-6 with Hilbert polynomial (6n−1)(g−1) by the group PGL5g−5.Example: A vector bundle W over an algebraic curve (or over a Riemann surface) is a stable vector bundle if andonly if

for all proper non-zero subbundles V of W and is semistable if this condition holds with < replaced by ≤.

References• Deligne, Pierre; Mumford, David (1969), "The irreducibility of the space of curves of given genus" [1],

Publications Mathématiques de l'IHÉS 36 (36): 75–109, doi:10.1007/BF02684599, MR0262240• Hilbert, D. (1893), "Über die vollen Invariantensysteme", Math. Annalen 42 (3): 313, doi:10.1007/BF01444162• Kirwan, Frances, Cohomology of quotients in symplectic and algebraic geometry. Mathematical Notes, 31.

Princeton University Press, Princeton, NJ, 1984. i+211 pp. MR0766741 ISBN 0-691-08370-3• Kraft, Hanspeter, Geometrische Methoden in der Invariantentheorie. (German) (Geometrical methods in invariant

theory) Aspects of Mathematics, D1. Friedr. Vieweg & Sohn, Braunschweig, 1984. x+308 pp. MR0768181 ISBN3-528-08525-8

Geometric invariant theory 23

• Mumford, David (1977), "Stability of projective varieties" [2], L'Enseignement Mathématique. RevueInternationale. IIe Série 23 (1): 39–110, ISSN 0013-8584, MR0450272

• Mumford, David; Fogarty, J.; Kirwan, F. (1994), Geometric invariant theory, Ergebnisse der Mathematik undihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], 34 (3rd ed.), Berlin, New York:Springer-Verlag, ISBN 978-3-540-56963-3, MR0214602( 1st ed 1965) MR0719371 (2nd ed) MR1304906(3rded.)

• E.B. Vinberg, V.L. Popov, Invariant theory, in Algebraic geometry. IV. Encyclopaedia of Mathematical Sciences,55 (translated from 1989 Russian edition) Springer-Verlag, Berlin, 1994. vi+284 pp. ISBN 3-540-54682-0

References[1] http:/ / www. numdam. org/ item?id=PMIHES_1969__36__75_0[2] http:/ / retro. seals. ch/ digbib/ view?rid=ensmat-001:1977:23::185

Gröbner basisIn computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basisis a particular kind of generating subset of an ideal I in a polynomial ring R. One can view it as a multivariate,non-linear generalization of:• the Euclidean algorithm for computation of univariate greatest common divisors,• Gaussian elimination for linear systems, and• integer programming problems.The theory of Gröbner bases for polynomial rings was developed by Bruno Buchberger in 1965, who named themafter his advisor Wolfgang Gröbner. The Association for Computing Machinery awarded him its 2007 ParisKanellakis Theory and Practice Award for this work. An analogous concept for local rings was developedindependently by Heisuke Hironaka in 1964, who named them standard bases. The analogous theory for free Liealgebras was developed by A. I. Shirshov in 1962 but his work remained largely unknown outside the Soviet Union.

Formal definitionA Gröbner basis G of an ideal I in a polynomial ring R over a field is characterised by any one of the followingproperties, stated relative to some monomial order:• the ideal given by the leading terms of polynomials in I is itself generated by the leading terms of the basis G;• the leading term of any polynomial in I is divisible by the leading term of some polynomial in the basis G;• multivariate division of any polynomial in the polynomial ring R by G gives a unique remainder;• multivariate division of any polynomial in the ideal I by G gives 0.All these properties are equivalent; different authors use different definitions depending on the topic they choose.The last two properties that allow calculations in the factor ring R/I with the same facility as modular arithmetic. It isa significant fact of commutative algebra that Gröbner bases always exist, and can be effectively obtained for anyideal starting with a generating subset.Multivariate division requires a monomial ordering, the basis depends on the monomial ordering chosen, anddifferent orderings can give rise to radically different Gröbner bases. Two of the most commonly used orderings arelexicographic ordering, and degree reverse lexicographic order (also called graded reverse lexicographic order orsimply total degree order). Lexicographic order eliminates variables, however the resulting Gröbner bases are oftenvery large and expensive to compute. Degree reverse lexicographic order typically provides for the fastest Gröbnerbasis computations. In this order monomials are compared first by total degree, with ties broken by taking thesmallest monomial with respect to lexicographic ordering with the variables reversed.

Gröbner basis 24

In most cases (polynomials in finitely many variables with complex coefficients or, more generally, coefficients overany field, for example), Gröbner bases exist for any monomial ordering. Buchberger's algorithm is the oldest andmost well-known method for computing them. Other methods are the Faugère F4 algorithm, based on the samemathematics as the Buchberger algorithm, and involutive approaches, based on ideas from differential algebra. [1]

There are also three algorithms for converting a Gröbner basis with respect to one monomial order to a Gröbner basiswith respect to a different monomial order: the FGLM algorithm, the Hilbert Driven Algorithm and the Gröbnerwalk algorithm. These algorithms are often employed to compute (difficult) lexicographic Gröbner bases from(easier) total degree Gröbner bases.A Gröbner basis is termed reduced if the leading coefficient of each element of the basis is 1 and no monomial inany element of the basis is in the ideal generated by the leading terms of the other elements of the basis. In the worstcase, computation of a Gröbner basis may require time that is exponential or even doubly exponential in the numberof solutions of the polynomial system (for degree reverse lexicographic order and lexicographic order, respectively).Despite these complexity bounds, both standard and reduced Gröbner bases are often computable in practice, andmost computer algebra systems contain routines to do so.The concept and algorithms of Gröbner bases have been generalized to modules over a polynomial ring, to freenon-commutative polynomial rings and, by Weispfenning and his school, to solvable polynomial rings such as Weylalgebras.

Properties and applications of Gröbner bases

Deciding equality of idealsReduced Gröbner bases can be shown to be unique for any given ideal and monomial ordering, and are also oftencomputable in practice. Thus one can determine if two ideals are equal by looking at their reduced Gröbner bases.

Deciding membership of idealsThe reduction of a polynomial f by the multivariate division algorithm for an ideal using a Gröbner basis will yield 0if and only if f is in the ideal. (By contrast, this is generally not true for a non-Gröbner basis with polynomials inmore than one variable). This gives a test for determining whether or not a polynomial is in an ideal with a given setof generators.

Elimination propertyIf a Gröbner basis for an ideal I in

k[x1, x2, ..., xn]is computed relative to the lexicographic ordering with

x1 > x2 > ... > xn,the intersection of I with

k[xk, xk+1, ..., xn]is given by the intersection of the Gröbner basis with

k[xk, xk+1, ..., xn].In particular a polynomial f lies in

k[xk, xk+1, ..., xn],if and only if its leading term lies in this subring. This is known as the elimination property.

Gröbner basis 25

Solving equationsIn particular, this gives us a method for solving simultaneous polynomial equations. If there are only finitely manysolutions (over an algebraic closure of the field in which the coefficients lie) to the system of equations

{f1[x1, ..., xn] = a1, ..., fm[x1, ..., xn] = am},we should be able to manipulate these equations to get something of the form

g(xn) = b.The elimination property says that if we compute a Gröbner basis for the ideal generated by {f1 – a1, ..., fm – am}relative to the right lexicographic ordering, then we can find the polynomial g as one of the elements of our basis.Furthermore, (taking k = n – 1) there will be another polynomial in the basis involving only xn – 1 and xn, so we cantake our possible solutions for xn and find corresponding values for xn – 1. This lifting continues all the way up untilwe've found the values of all the variables.

Conversion of parametric equationsThe same elimination property can almost be used to convert parametric equations of polynomials intononparametric equations. Given the equations

{x1 = f1(t1, ..., tm), ..., xn = fn(t1, ..., tm)},we compute a Gröbner basis for the ideal generated by

{x1 – f1, ..., xn – fn}relative to any ordering that places polynomials involving t greater than those that don't: for example, lexicographicordering with

t1 > t2 > ... > tm > x1 > ... > xn.Taking only the elements of the basis that do not involve the t variables, we get a set of equations describing not theoriginal surface, but the smallest affine variety containing it.

Intersecting ideals• If I is generated by

{f1, ..., fm}and J is generated by some

{g1, ..., gk},then the intersection of I and J can be found by taking a Gröbner basis for the ideal generated by

{tf1, ..., tfm, (1 – t)g1, ..., (1 – t)gk}relative to any lexicographic ordering that places t first, then taking only those terms not involving t. In particular,this allows us to calculate the least common multiple (and hence the greatest common divisor) of two polynomials fand g, since it is the generator of the intersection of the ideals generated by f and by g. This is true even if we do notknow how to factor the polynomials! Also, note that for more than one variable the polynomial ring is not aEuclidean domain, so the Euclidean algorithm doesn't work here.

Gröbner basis 26

References[1] Vladimir P. Gerdt, Yuri A. Blinkov (1998). Involutive Bases of Polynomial Ideals, Mathematics and Computers in Simulation, 45:519ff

Further reading• William W. Adams, Philippe Loustaunau (1994). An Introduction to Gröbner Bases. American Mathematical

Society, Graduate Studies in Mathematics, Volume 3. ISBN 0-8218-3804-0• Thomas Becker, Volker Weispfenning (1998). Gröbner Bases. Springer Graduate Texts in Mathematics 141.

ISBN 0-387-97971-7• Bruno Buchberger (1965). An Algorithm for Finding the Basis Elements of the Residue Class Ring of a Zero

Dimensional Polynomial Ideal (http:/ / www. ricam. oeaw. ac. at/ Groebner-Bases-Bibliography/ gbbib_files/publication_706. pdf). Ph.D. dissertation, University of Innsbruck. English translation by Michael Abramson inJournal of Symbolic Computation 41 (2006): 471–511. [This is Buchberger's thesis inventing Gröbner bases.]

• Bruno Buchberger (1970). An Algorithmic Criterion for the Solvability of a System of Algebraic Equations (http:// www. ricam. oeaw. ac. at/ Groebner-Bases-Bibliography/ gbbib_files/ publication_699. pdf). AequationesMathematicae 4 (1970): 374–383. English translation by Michael Abramson and Robert Lumbert in GröbnerBases and Applications (B. Buchberger, F. Winkler, eds.). London Mathematical Society Lecture Note Series251, Cambridge University Press, 1998, 535–545. ISBN 0-521-63298-6 (This is the journal publication ofBuchberger's thesis.)

• David Cox, John Little, and Donal O'Shea (1997). "Chapter 2: Gröbner Bases". Ideals, Varieties, and Algorithms:An Introduction to Computational Algebraic Geometry and Commutative Algebra. Springer.ISBN 0-387-94680-2.

• Ralf Fröberg (1997). An Introduction to Gröbner Bases. Wiley & Sons. ISBN 0-471-97442-0.• Sturmfels, Bernd (November 2005), "What is . . . a Gröbner Basis?" (http:/ / math. berkeley. edu/ ~bernd/ what-is.

pdf), Notices of the American Mathematical Society 52 (10): 1199–1200, a brief introduction.• A. I. Shirshov (1999). "Certain algorithmic problems for Lie algebras" (http:/ / www. ricam. oeaw. ac. at/

Groebner-Bases-Bibliography/ gbbib_files/ publication_835. pdf). ACM SIGSAM Bulletin 33 (2): 3–6. (translatedfrom Sibirsk. Mat. Zh. Siberian Mathemaics Journal, 3 (1962), 292–296)

• M. Aschenbrenner and C. Hillar, Finite generation of symmetric ideals (http:/ / www. ams. org/ tran/2007-359-11/ S0002-9947-07-04116-5/ home. html), Trans. Amer. Math. Soc. 359 (2007), 5171–5192 (oninfinite dimensional Gröbner bases for polynomial rings in infinitely many indeterminates).

External links• B. Buchberger, Groebner Bases: A Short Introduction for Systems Theorists (http:/ / www. risc. uni-linz. ac. at/

people/ buchberg/ papers/ 2001-02-19-A. pdf) in Proceedings of EUROCAST 2001.• Buchberger, B. and Zapletal, A. Gröbner Bases Bibliography. (http:/ / www. ricam. oeaw. ac. at/

Groebner-Bases-Bibliography/ search. php)• Comparative Timings Page for Gröbner Bases Software (http:/ / magma. maths. usyd. edu. au/ users/ allan/ gb)• ogb (http:/ / grobner. nuigalway. ie/ grobner/ basis. html) Online Gröbner Basis, Galway, Éire• Java applet for computing Gröbner bases (http:/ / www. iisdavinci. it/ jeometry/ jeometry_groebner. html) by

Fabrizio• Gröbner Basis Theory (http:/ / www. cs. le. ac. uk/ people/ ah83/ grobner/ ) Leicester University• Prof. Bruno Buchberger (http:/ / www. risc. jku. at/ people/ buchberg/ ) Bruno Buchberger• Weisstein, Eric W., " Gröbner Basis (http:/ / mathworld. wolfram. com/ GroebnerBasis. html)" from MathWorld.

Haboush's theorem 27

Haboush's theoremIn mathematics Haboush's theorem, often still referred to as the Mumford conjecture, states that for anysemisimple algebraic group G over a field K, and for any linear representation ρ of G on a K-vector space V, givenv ≠ 0 in V that is fixed by the action of G, there is a G-invariant polynomial F on V such that

F(v) ≠ 0.The polynomial can be taken to be homogeneous, in other words an element of a symmetric power of the dual of V,and if the characteristic is p>0 the degree of the polynomial can be taken to be a power of p. When K hascharacteristic 0 this was well known; in fact Weyl's theorem on the complete reducibility of the representations of Gimplies that F can even be taken to be linear. Mumford's conjecture about the extension to prime characteristic p wasproved by W. J. Haboush (1975), about a decade after the problem had been posed by David Mumford, in theintroduction to the first edition of his book Geometric Invariant Theory.

ApplicationsHaboush's theorem can be used to generalize results of geometric invariant theory from characteristic 0, where theywere already known, to characteristic p>0. In particular Nagata's earlier results together with Haboush's theoremshow that if a reductive group (over an algebraically closed field) acts on a finitely generated algebra then the fixedsubalgebra is also finitely generated.Haboush's theorem implies that if G is a reductive algebraic group acting regularly on an affine algebraic variety,then disjoint closed invariant sets X and Y can be separated by an invariant function f (this means that f is 0 on X and1 on Y).C.S. Seshadri (1977) extended Haboush's theorem to reductive groups over schemes.It follows from the work of Nagata (1963), Haboush, and Popov that the following conditions are equivalent for anaffine algebraic group G over a field K:• G is reductive (its unipotent radical is trivial).• For any non-zero invariant vector in a rational representation on G, there is an invariant homogeneous polynomial

that does not vanish on it.• For any finitely generated K algebra acted on rationally by G, the algebra of fixed elements is finitely generated.

ProofThe theorem is proved in several steps as follows:• We can assume that the group is defined over an algebraically closed field K of characteristic p>0.• Finite groups are easy to deal with as one can just take a product over all elements, so one can reduce to the case

of connected reductive groups (as the connected component has finite index). By taking a central extensionwhich is harmless one can also assume the group G is simply connected.

• Let A(G) be the coordinate ring of G. This is a representation of G with G acting by left translations. Pick anelement v′ of the dual of V that has value 1 on the invariant vector v. The map V to A(G) by sending w∈V to theelement a∈A(G) with a(g) = v′(g(w)). This sends v to 1∈A(G), so we can assume that V⊂A(G) and v=1.

• The structure of the representation A(G) is given as follows. Pick a maximal torus T of G, and let it act on A(G) byright translations (so that it commutes with the action of G). Then A(G) splits as a sum over characters λ of T ofthe subrepresentations A(G)λ of elements transforming according to λ. So we can assume that V is contained in theT-invariant subspace A(G)λ of A(G).

• The representation A(G)λ is an increasing union of subrepresentations of the form Eλ+nρ⊗Enρ, where ρ is the Weyl vector for a choice of simple roots of T, n is a positive integer, and Eμ is the space of sections of the line

Haboush's theorem 28

bundle over G/B corresponding to a character μ of T, where B is a Borel subgroup containing T.• If n is sufficiently large then Enρ has dimension (n+1)N where N is the number of positive roots. This is because in

characteristic 0 the corresponding module has this dimension by the Weyl character formula, and for n largeenough that the line bundle over G/B is very ample, Enρ has the same dimension as in characteristic 0.

• If q=pr for a positive integer r, and n=q−1, then Enρ contains the Steinberg representation of G(Fq) of dimensionqN. (Here Fq ⊂ K is the finite field of order q.) The Steinberg representation is an irreducible representation ofG(Fq) and therefore of G(K), and for r large enough it has the same dimension as Enρ, so there are infinitely manyvalues of n such that Enρ is irreducible.

• If Enρ is irreducible it is isomorphic to its dual, so Enρ⊗Enρ is isomorphic to End(Enρ). Therefore the T-invariantsubspace A(G)λ of A(G) is an increasing union of subrepresentations of the form End(E) for representations E (ofthe form E(q−1)ρ)). However for representations of the form End(E) an invariant polynomial that separates 0 and 1is given by the determinant. This completes the sketch of the proof of Haboush's theorem.

References• Demazure, Michel (1976), "Démonstration de la conjecture de Mumford (d'après W. Haboush)", Séminaire

Bourbaki (1974/1975: Exposés Nos. 453--470), Lecture Notes in Math., 514, Berlin: Springer, pp. 138–144,doi:10.1007/BFb0080063, ISBN 978-3-540-07686-5, MR0444786

• Haboush, W. J. (1975), "Reductive groups are geometrically reductive", Ann. Of Math. (The Annals ofMathematics, Vol. 102, No. 1) 102 (1): 67–83, doi:10.2307/1970974, JSTOR 1970974

• Mumford, D.; Fogarty, J.; Kirwan, F. Geometric invariant theory. Third edition. Ergebnisse der Mathematik undihrer Grenzgebiete (2) (Results in Mathematics and Related Areas (2)), 34. Springer-Verlag, Berlin, 1994.xiv+292 pp. MR1304906 ISBN 3-540-56963-4

• Nagata, Masayoshi (1963), "Invariants of a group in an affine ring" [1], Journal of Mathematics of KyotoUniversity 3: 369–377, ISSN 0023-608X, MR0179268

• M. Nagata, T. Miyata, "Note on semi-reductive groups" J. Math. Kyoto Univ. , 3 (1964) pp. 379–382• Popov, V.L. (2001), "Mumford hypothesis" [2], in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,

ISBN 978-1556080104• C.S. Seshadri, "Geometric reductivity over arbitrary base" Adv. Math. , 26 (1977) pp. 225–274

References[1] http:/ / projecteuclid. org/ euclid. kjm/ 1250524787[2] http:/ / www. encyclopediaofmath. org/ index. php?title=M/ m065570

Hall algebra 29

Hall algebraIn mathematics the Hall algebra is an associative algebra with a basis corresponding to isomorphism classes offinite abelian p-groups. It was first discussed by E. Steinitz (1901) but forgotten until it was rediscovered by PhilipHall (1959), both of whom published no more than brief summaries of their work. The Hall polynomials are thestructure constants of the Hall algebra. The Hall algebra plays an important role in the theory ofKashiwara–Lusztig's canonical bases in quantum groups. Ringel (1990) generalized Hall algebras to more generalcategories, such as the category of representations of a quiver.

ConstructionA finite abelian p-group M is a direct sum of cyclic p-power components where is apartition of called the type of M. Let be the number of subgroups N of M such that N has type and the

quotient M/N has type . Hall proved that the functions g are polynomial functions of p with integer coefficients.Thus we may replace p with an indeterminate q, which results in the Hall polynomials

Hall next constructs an associative ring over , now called the Hall algebra. This ring has a basis consistingof the symbols and the structure constants of the multiplication in this basis are given by the Hall polynomials:

It turns out that H is a commutative ring, freely generated by the elements corresponding to the elementaryp-groups. The linear map from H to the algebra of symmetric functions defined on the generators by the formula

(where en is the nth elementary symmetric function) uniquely extends to a ring homomorphism and the images of thebasis elements may be interpreted via the Hall–Littlewood symmetric functions. Specializing q to 1, thesesymmetric functions become Schur functions, which are thus closely connected with the theory of Hall polynomials.

References• Hall, Philip (1959), "The algebra of partitions", Proceedings of the 4th Canadian mathematical congress, Banff,

pp. 147–159• George Lusztig, Quivers, perverse sheaves, and quantized enveloping algebras. J. Amer. Math. Soc. 4 (1991), no.

2, 365–421.• Macdonald, I. G. (1995), Symmetric functions and Hall polynomials [1], Oxford Mathematical Monographs (2nd

ed.), The Clarendon Press Oxford University Press, ISBN 978-0-19-853489-1, MR1354144• Ringel, Claus Michael (1990), "Hall algebras and quantum groups", Inventiones Mathematicae 101 (3): 583–591,

doi:10.1007/BF01231516, MR1062796• Schiffmann, O (2006), Lectures on Hall algebras, arXiv:math/0611617• Steinitz, E. (1901), "Zur Theorie der Abel'schen Gruppen", Jahresbericht der DMV 9: 80–85

References[1] http:/ / www. oup. com/ uk/ catalogue/ ?ci=9780198504504

Hilbert's basis theorem 30

Hilbert's basis theoremIn mathematics, specifically commutative algebra, Hilbert's basis theorem states that every ideal in the ring ofmultivariate polynomials over a Noetherian ring is finitely generated. This can be translated into algebraic geometryas follows: every algebraic set over a field can be described as the set of common roots of finitely many polynomialequations. Hilbert (1890) proved the theorem (for the special case of polynomial rings over a field) in the course ofhis proof of finite generation of rings of invariants.Hilbert produced an innovative proof by contradiction using mathematical induction; his method does not give analgorithm to produce the finitely many basis polynomials for a given ideal: it only shows that they must exist. Onecan determine basis polynomials using the method of Gröbner bases.

ProofThe following more general statement will be proved.

Theorem. If is a left- (respectively right-) Noetherian ring, then the polynomial ring is also a left-(respectively right-) Noetherian ring.It suffices to consider just the "Left" case.Proof (Theorem)

Suppose per contra that were a non-finitely generated left-ideal. Then it would be that by recursion(using the countable axiom of choice) that a sequence of polynomials could be found so that, letting

of minimal degree. It is clear that is a non-decreasingsequence of naturals. Now consider the left-ideal over where the are the leadingcoefficients of the . Since is left-Noetherian, we have that must be finitely generated; and since the comprise an -basis, it follows that for a finite amount of them, say will suffice. So for example,

some Now consider whose leadingterm is equal to that of moreover so of degree contradicting minimality. (Thm)

A constructive proof (not invoking the axiom of choice) also exists.Proof (Theorem):

Let be a left-ideal. Let be the set of leading coefficients of members of This is obviously aleft-ideal over and so is finitely generated by the leading coefficients of finitely many members of say

Let Let be the set of leading coefficients of members of whose degree

is As before, the are left-ideals over and so are finitely generated by the leading coefficients of finitelymany members of say with degrees Now let be the left-ideal generated

by We have and claim also Suppose per contra this were not so. Then let be of minimal degree, and denote its leading coefficientby Case 1: Regardless of this condition, we have so is a left-linear combination of the coefficients of the Consider which has the same leading term as

moreover so of degree contradicting minimality.

Hilbert's basis theorem 31

Case 2: Then so is a left-linear combination of the leading

coefficients of the Considering we yield a similar contradiction as in

Case 1.Thus our claim holds, and which is finitely generated.

(Thm)

Note that the only reason we had to split into two cases was to ensure that the powers of multiplying the factors,were non-negative in the constructions.

ApplicationsLet be a Nötherian commutative ring. Hilbert's basis theorem has some immediate corollaries. First, byinduction we see that will also be Nötherian. Second, since any affine variety over (i.e. a locus-set of a collection of polynomials) may be written as the locus of an ideal

and further as the locus of its generators, it follows that every affine variety is thelocus of finitely many polynomials — i.e. the intersection of finitely many hypersurfaces. Finally, if were afinitely-generated -algebra, then we know that (i.e. mod-ing out byrelations), where a set of polynomials. We can assume that is an ideal and thus is finitely generated. So would be a free -algebra (on generators) generated by finitely many relations

.

Mizar SystemThe Mizar project has completely formalized and automatically checked a proof of Hilbert's basis theorem in theHILBASIS file [1].

References• Cox, Little, and O'Shea, Ideals, Varieties, and Algorithms, Springer-Verlag, 1997.• Hilbert, David (1890), "Ueber die Theorie der algebraischen Formen", Mathematische Annalen 36 (4): 473–534,

doi:10.1007/BF01208503, ISSN 0025-5831

References[1] http:/ / www. mizar. org/ JFM/ Vol12/ hilbasis. html

Hilbert's fourteenth problem 32

Hilbert's fourteenth problemIn mathematics, Hilbert's fourteenth problem, that is, number 14 of Hilbert's problems proposed in 1900, askswhether certain rings are finitely generated.The setting is as follows: Assume that k is a field and let K be a subfield of the field of rational functions in nvariables,

k(x1, ..., xn ) over k.Consider now the ring R defined as the intersection

Hilbert conjectured that all such subrings are finitely generated. It can be shown that the field K is always finitelygenerated as a field, in other words, there exist finitely many elements

yi, i = 1 ,...,d in Ksuch that every element in R can be rationally represented by the yi. But this does not imply that the ring R is finitelygenerated as a ring, even if all the elements yi could be chosen from R.After some results were obtained confirming Hilbert's conjecture in special cases and for certain classes of rings (inparticular the conjecture was proved unconditionally for n = 1 and n = 2 by Zariski in 1954) then in 1959 MasayoshiNagata found a counterexample to Hilbert's conjecture. The counterexample of Nagata is a suitably constructed ringof invariants for the action of a linear algebraic group.

HistoryThe problem originally arose in algebraic invariant theory. Here the ring R is given as a (suitably defined) ring ofpolynomial invariants of a linear algebraic group over a field k acting algebraically on a polynomial ring k[x1, ..., xn](or more generally, on a finitely generated algebra defined over a field). In this situation the field K is the field ofrational functions (quotients of polynomials) in the variables xi which are invariant under the given action of thealgebraic group, the ring R is the ring of polynomials which are invariant under the action. A classical example innineteenth century was the extensive study (in particular by Cayley, Sylvester, Clebsch, Paul Gordan and alsoHilbert) of invariants of binary forms in two variables with the natural action of the special linear group SL2(k) on it.Hilbert himself proved the finite generation of invariant rings in the case of the field of complex numbers for someclassical semi-simple Lie groups (in particular the general linear group over the complex numbers) and specificlinear actions on polynomial rings, i.e. actions coming from finite-dimensional representations of the Lie-group. Thisfiniteness result was later extended by Hermann Weyl to the class of all semi-simple Lie-groups. A major ingredientin Hilbert's proof is the Hilbert basis theorem applied to the ideal inside the polynomial ring generated by theinvariants.

Zariski's formulationZariski's formulation of Hilbert's fourteenth problem asks whether, for a quasi-affine algebraic variety X over a fieldk, possibly assuming X normal or smooth, the ring of regular functions on X is finitely generated over k.Zariski's formulation was shown[1] to be equivalent to the original problem, for X normal.

Nagata's counterexampleNagata (1958) gave the following counterexample to Hilbert's problem. The field k is a field containing 48 elements a1i, ...,a16i, for i=1, 2, 3 that are algebraically independent over the prime field. The ring R is the polynomial ring k[x1,...,x16, t1,...,t16] in 32 variables. The vector space V is a 13 dimensional vector space over k consisting of all

Hilbert's fourteenth problem 33

vectors ( b1,...,b16) in k16 orthogonal to the each of the three vectors (a1i, ...,a16i) for i=1, 2, 3. The vector space V is a13-dimensional commutative unipotent algebraic group under addition, and its elements act on R by fixing allelements tj and taking xj to xj + bjtj. Then the ring of elements of R invariant under the action of the group V is not afinitely generated k-algebra.Several authors have reduced the sizes of the group and the vector space in Nagata's example. For example, Totaro(2008) showed that over any field there is an action of the sum G of 3 copies of the additive group on k18 whosering of invariants is not finitely generated.

References• Nagata, Masayoshi (1960), "On the fourteenth problem of Hilbert" [2], Proc. Internat. Congress Math. 1958,

Cambridge University Press, pp. 459–462, MR0116056• M. Nagata: Lectures on the fourteenth problem of Hilbert. Lect. Notes 31, Tata Inst. Bombay, 1965.• Totaro, Burt (2008), "Hilbert's 14th problem over finite fields and a conjecture on the cone of curves", Compositio

Mathematica 144 (5): 1176–1198, doi:10.1112/S0010437X08003667, ISSN 0010-437X, MR2457523• O. Zariski, Interpretations algebrico-geometriques du quatorzieme probleme de Hilbert, Bulletin des Sciences

Mathematiques 78 (1954), pp. 155–168.[1] Winkelmann, Jörg (2003), "Invariant rings and quasiaffine quotients", Math. Z. 244 (1): 163–174, doi:10.1007/s00209-002-0484-9.[2] http:/ / mathunion. org/ ICM/ ICM1958/

Hilbert's syzygy theoremIn mathematics, Hilbert's syzygy theorem is a result of commutative algebra, first proved by David Hilbert (1890)in connection with the syzygy (relation) problem of invariant theory. Roughly speaking, starting with relationsbetween polynomial invariants, then relations between the relations, and so on, it explains how far one has to go toreach a clarified situation. It is now considered to be an early result of homological algebra, and through the depthconcept, to be a measure of the non-singularity of affine space.

Formal statementA contemporary formal statement is the following. Let k be a field and M a finitely generated module over thepolynomial ring

Hilbert's syzygy theorem then states that there exists a free resolution of M of length at most n.

References• David Eisenbud, Commutative algebra. With a view toward algebraic geometry. Graduate Texts in Mathematics,

150. Springer-Verlag, New York, 1995. xvi+785 pp. ISBN 0-387-94268-8; ISBN 0-387-94269-6 MR1322960

Hodge bundle 34

Hodge bundleIn mathematics, the Hodge bundle, named after W. V. D. Hodge, appears in the study of families of curves, where itprovides an invariant in the moduli theory of algebraic curves. Furthermore, it has applications to the theory ofmodular forms on reductive algebraic groups[1] and string theory.[2]

Let be the moduli space of algebraic curves of genus g curves over some scheme. The Hodge bundle Λg is avector bundle on whose fiber at a point C in is the space of holomorphic differentials on the curve C. Todefine the Hodge bundle, let be the universal algebraic curve of genus g and let ωg be its relativedualizing sheaf. The Hodge bundle is the pushforward of this sheaf, i.e.[3]

Notes[1] van der Geer 2008, §13[2] Liu 2006, §5[3] Harris & Morrison 1998, p. 155

References• van der Geer, Gerard (2008), "Siegel modular forms and their applications", in Ranestad, Kristian, The 1-2-3 of

modular forms, Universitext, Berlin: Springer-Verlag, pp. 181–245, doi:10.1007/978-3-540-74119-0,ISBN 978-3-540-74117-6, MR2409679

• Harris, Joe; Morrison, Ian (1998), Moduli of curves, Graduate Texts in Mathematics, 187, Springer-Verlag,ISBN 978-0-387-98429-2, MR1631825

• Liu, Kefeng (2006), "Localization and conjectures from string duality", in Ge, Mo-Lin; Zhang, Weiping,Differential geometry and physics, Nankai Tracts in Mathematics, 10, World Scientific, pp. 63–105,ISBN 978-9-812-70377-4, MR2322389

Invariant estimator 35

Invariant estimatorIn statistics, the concept of being an invariant estimator is a criterion that can be used to compare the properties ofdifferent estimators for the same quantity. It is a way of formalising the idea that an estimator should have certainintuitively appealing qualities. Strictly speaking, "invariant" would mean that the estimates themselves areunchanged when both the measurements and the parameters are transformed in a compatible way, but the meaninghas been extended to allow the estimates to change in appropriate ways with such transformations. The termequivariant estimator is used in formal mathematical contexts that include a precise description of the relation ofthe way the estimator changes in response to changes to the dataset and parameterisation: this corresponds to the useof "equivariance" in more general mathematics.

General setting

BackgroundIn statistical inference, there are several approaches to estimation theory that can be used to decide immediately whatestimators should be used according to those approaches. For example, ideas from Bayesian inference would leaddirectly to Bayesian estimators. Similarly, the theory of classical statistical inference can sometimes lead to strongconclusions about what estimator should be used. However, the usefulness of these theories depends on having afully prescribed statistical model and may also depend on having a relevant loss function to determine the estimator.Thus a Bayesian analysis might be undertaken, leading to a posterior distribution for relevant parameters, but the useof a specific utility or loss function may be unclear. Ideas of invariance can then be applied to the task ofsummarising the posterior distribution. In other cases, statistical analyses are undertaken without a fully definedstatistical model or the classical theory of statistical inference cannot be readily applied because the family of modelsbeing considered are not amenable to such treatment. In addition to these cases where general theory does notprescribe an estimator, the concept of invariance of an estimator can be applied when seeking estimators ofalternative forms, either for the sake of simplicity of application of the estimator or so that the estimator is robust.The concept of invariance is sometimes used on its own as a way of choosing between estimators, but this is notnecessarily definitive. For example, a requirement of invariance may be incompatible with the requirement that theestimator be mean-unbiased; on the other hand, the criterion of median-unbiasedness is defined in terms of theestimator's sampling distribution and so is invariant under many transformations.One use of the concept of invariance is where a class or family of estimators is proposed and a particular formulationmust be selected amongst these. One procedure is to impose relevant invariance properties and then to find theformulation within this class that has the best properties, leading to what is called the optimal invariant estimator.

Some classes of invariant estimatorsThere are several types of transformations that are usefully considered when dealing with invariant estimators. Eachgives rise to a class of estimators which are invariant to those particular types of transformation.• Shift invariance: Notionally, estimates of a location parameter should be invariant to simple shifts of the data

values. If all data values are increased by a given amount, the estimate should change by the same amount. Whenconsidering estimation using a weighted average, this invariance requirement immediately implies that theweights should sum to one. While the same result is often derived from a requirement for unbiasedness, the use of"invariance" does not require that a mean value exists and makes no use of any probability distribution at all.

• Scale invariance: Note that this is a topic not directly covered in scale invariance.• Parameter-transformation invariance: Here, the transformation applies to the parameters alone. The concept here

is that essentially the same inference should be made from data and a model involving a parameter θ as would be

Invariant estimator 36

made from the same data if the model used a parameter φ, where φ is a one-to-one transformation of θ, φ=h(θ).According to this type of invariance, results from transformation-invariant estimators should also be related byφ=h(θ). Maximum likelihood estimators have this property.

• Permutation invariance: Where a set of data values can be represented by a statistical model that they areoutcomes from independent and identically distributed random variables, it is reasonable to impose therequirement that any estimator of any property of the common distribution should be permutation-invariant:specifically that the estimator, considered as a function of the set of data-values, should not change if items ofdata are swapped within the dataset.

The combination of permutation invariance and location invariance for estimating a location parameter from anindependent and identically distributed dataset using a weighted average implies that the weights should be identicaland sum to one. Of course, estimators other than a weighted average may be preferable.

Optimal invariant estimatorsUnder this setting, we are given a set of measurements which contains information about an unknown parameter

. The measurements are modelled as a vector random variable having a probability density function which depends on a parameter vector .The problem is to estimate given . The estimate, denoted by , is a function of the measurements and belongsto a set . The quality of the result is defined by a loss function which determines a risk function

. The sets of possible values of , , and are denoted by , , and ,respectively.

Mathematical setting

DefinitionAn invariant estimator is an estimator which obeys the following two rules:1. Principle of Rational Invariance: The action taken in a decision problem should not depend on transformation on

the measurement used2. Invariance Principle: If two decision problems have the same formal structure (in terms of , , and

), then the same decision rule should be used in each problem.To define an invariant or equivariant estimator formally, some definitions related to groups of transformations areneeded first. Let denote the set of possible data-samples. A group of transformations of , to be denoted by

, is a set of (measurable) 1:1 and onto transformations of into itself, which satisfies the following conditions:

1. If and then 2. If then , where (That is, each transformation has an inverse within the

group.)3. (i.e. there is an identity transformation )Datasets and in are equivalent if for some . All the equivalent points form anequivalence class. Such an equivalence class is called an orbit (in ). The orbit, , is the set

. If consists of a single orbit then is said to be transitive.A family of densities is said to be invariant under the group if, for every and there exists aunique such that has density . will be denoted .If is invariant under the group then the loss function is said to be invariant under if for every

and there exists an such that for all . The transformedvalue will be denoted by .

Invariant estimator 37

In the above, is a group of transformations from to itself and is agroup of transformations from to itself.An estimation problem is invariant(equivariant) under if there exist three groups as defined above.For an estimation problem that is invariant under , estimator is an invariant estimator under if, for all

and ,

Properties1. The risk function of an invariant estimator, , is constant on orbits of . Equivalently

for all and .2. The risk function of an invariant estimator with transitive is constant.For a given problem, the invariant estimator with the lowest risk is termed the "best invariant estimator". Bestinvariant estimator cannot always be achieved. A special case for which it can be achieved is the case when istransitive.

Example: Location parameter

Suppose is a location parameter if the density of is of the form . For and, the problem is invariant under . The invariant

estimator in this case must satisfy

thus it is of the form ( ). is transitive on so the risk does not vary with : that is,. The best invariant estimator is the one that brings the risk

to minimum.In the case that L is the squared error

Pitman estimator

The estimation problem is that has density , where θ is aparameter to be estimated, and where the loss function is . This problem is invariant with the following(additive) transformation groups:

The best invariant estimator is the one that minimizes

and this is Pitman's estimator (1939).For the squared error loss case, the result is

If (i.e. a multivariate normal distribution with independent, unit-variance components) then

Invariant estimator 38

If (independent components having a Cauchy distribution with scale parameter σ) then,. However the result is

with

References• Berger, James O. (1985). Statistical decision theory and Bayesian Analysis (2nd ed.). New York:

Springer-Verlag. ISBN 0-387-96098-8. MR0804611.• Freue, Gabriela V. Cohen (2007) "The Pitman estimator of the Cauchy location parameter", Journal of Statistical

Planning and Inference, 137, 1900–1913 doi:10.1016/j.jspi.2006.05.002• Pitman, E.J.G. (1939) "The estimation of the location and scale parameters of a continuous population of any

given form", Biometrika, 30 (3/4), 391–421. JSTOR 2332656• Pitman, E.J.G. (1939) "Tests of Hypotheses Concerning Location and Scale Parameters", Biometrika, 31 (1/2),

200–215. JSTOR 2334983

Invariant of a binary formIn mathematical invariant theory, an invariant of a binary form is a polynomial in the coefficients of a binary formin two variables x and y that remains invariant under unimodular transformations of the variables x and y.

TerminologyA binary form (of degree n) is a homogeneous polynomial Σ ( )an−ix

n−iyi = anxn + ( )an−1xn−1y + ...+ a0yn. The group SL2(C) acts on these forms by taking x to ax + by and y to cx + dy. This induces an action on thespace spanned by a0, ..., an and on the polynomials in these variables. An invariant is a polynomial in these n + 1variables a0, ..., an that is invariant under this action. More generally a covariant is a polynomial in a0, ..., an, x, ythat is invariant, so an invariant is a special case of a covariant where the variables x and y do not occur. Moregenerally still, a simultaneous invariant is a polynomial in the coefficients of several different forms in x and y.In terms of representation theory, given any representation V of the group SL2(C) one can ask for the ring ofinvariant polynomials on V. Invariants of a binary form of degree n correspond to taking V to be the(n + 1)-dimensional irreducible representation, and covariants correspond to taking V to be the sum of the irreduciblerepresentations of dimensions 2 and n + 1.The invariants of a binary form are a graded algebra, and Gordan (1868) proved that this algebra is finitely generatedif the base field is the complex numbers.Forms of degrees 2, 3, 4, 5, 6, 7, 8, 9, 10 are sometimes called quadrics, cubic, quartics, quintics, sextics, septics,octavics, nonics, and decimics. "Quantic" is an old name for a form of arbitrary degree. Forms in 1, 2, 3, ... variablesare called unary, binary, ternary, ... forms.

Invariant of a binary form 39

ExamplesA form f is itself is a covariant of degree 1 and order n.The discriminant of a form is an invariant.The resultant of two forms is a simultaneous invariant of them.The Hessian covariant of a form Hilbert (1993, p.88) is the determinant of the Hessian matrix

It is a covariant of order 2n− 4 and degree 2.The catalecticant is an invariant of a form of even degree.The Jacobian

is a simultaneous invariant of two forms f, g.

The ring of invariantsThe structure of the ring of invariants has been worked out for small degrees. Sylvester & Franklin (1879) gavetables of the numbers of generators of invariants and covariants for forms of degree up to 10.1. For linear forms ax + by the only invariants are constants. The algebra of covariants is generated by the form

itself of degree 1 and order 1.2. The algebra of invariants of the quadratic form ax2 + 2bxy + cy2 is a polynomial algebra in 1 variable generated

by the discriminant b2 − ac of degree 2. The algebra of covariants is a polynomial algebra in 2 variables generatedby the discriminant together with the form f itself (of degree 1 and order 2). (Schur 1968, II.8) (Hilbert 1993,XVI, XX)

3. The algebra of invariants of the cubic form ax3 + 3bx2y + 3cxy2 + dy3 is a polynomial algebra in 1 variablegenerated by the discriminant D = 3b2c2 + 6abcd − 4b3d − 4c3a − a2d2 of degree 4. The algebra of covariants isgenerated by the discriminant, the form itself (degree 1, order 3), the Hessian H (degree 2, order 2) and acovariant T of degree 3 and order 3. They are related by the syzygy 4h3=Df2-T2 of degree 6 and order 6. (Schur1968, II.8) (Hilbert 1993, XVII, XX)

4. The algebra of invariants of a quartic form is generated by invariants i, j of degrees 2, 3. The algebra ofcovariants is generated by these two invariants together with the form f of degree 1 and order 4, the Hessian H ofdegree 2 and order 4, and a covariant T of degree 3 and order 6. They are related by a syzygy jf3−Hf2i+4H3+T2=0of degree 6 and order 12. (Schur 1968, II.8) (Hilbert 1993, XVIII, XXII)

5. The algebra of invariants of a quintic form was found by Sylvester and is generated by invariants of degree 4, 8,12, 18. The generators of degrees 4, 8, 12 generate a polynomial ring, which contains the square of the generatorof degree 18. The invariants are rather complicated to write out explicitly: Sylvester showed that the generators ofdegrees 4, 8, 12, 18 have 12, 59, 228, and 848 terms often with very large coefficients. (Schur 1968, II.9) (Hilbert1993, XVIII)

6. The algebra of invariants of a sextic form is generated by invariants of degree 2, 4, 6, 10, 15. The generators ofdegrees 2, 4, 6, 10 generate a polynomial ring, which contains the square of the generator of degree 15. (Schur1968, II.9)

7. von Gall (1888) and Dixmier & Lazard (1986) showed that the algebra of invariants of a degree 7 form is generated by a set with 1 invariant of degree 4, 3 of degree 8, 6 of degree 12, 4 of degree 14, 2 of degree 16, 9 of

Invariant of a binary form 40

degree 18, and one of each of the degrees 20, 22, 26, 308. von Gall (1880) and Shioda (1967) showed that the algebra of invariants of a degree 8 form is generated by 9

invariants of degrees 2, 3, 4, 5, 6, 7, 8, 9, 10, and the ideal of relations between them is generated by elements ofdegrees 16, 17, 18, 19, 20.

9. Brouwer & Popoviciu (2010a) showed that the algebra of invariants of a degree 9 form is generated by 92invariants

10. Brouwer & Popoviciu (2010b) showed that the algebra of invariants of a degree 10 form is generated by 106invariants

The number of generators for invariants and covariants of binary forms can be found in (sequence A036983 inOEIS) and (sequence A036984 in OEIS), respectively.

Invariants of a ternary cubicThe algebra of invariants of a ternary cubic under SL3(C) is a polynomial algebra generated by two invariants ofdegrees 4 and 6. The invariants are rather complicated, and are given explicitly in (Sturmfels 1993, 4.4.7, 4.5.3)

References• Brouwer, Andries E.; Popoviciu, Mihaela (2010a), "The invariants of the binary nonic", Journal of Symbolic

Computation 45 (6): 709–720, doi:10.1016/j.jsc.2010.03.003, ISSN 0747-7171, MR2639312• Brouwer, Andries E.; Popoviciu, Mihaela (2010b), "The invariants of the binary decimic", Journal of Symbolic

Computation 45 (8): 837–843, doi:10.1016/j.jsc.2010.03.002, ISSN 0747-7171, MR2657667• Dixmier, Jacques; Lazard, D. (1988), "Minimum number of fundamental invariants for the binary form of degree

7", Journal of Symbolic Computation 6 (1): 113–115, doi:10.1016/S0747-7171(88)80026-9, ISSN 0747-7171,MR961375

• Gall, F. von (1880), "Das vollständige Formensystem einer binären Form achter Ordnung", MathematischeAnnalen 17 (1): 31–51, doi:10.1007/BF01444117, ISSN 0025-5831, MR1510048

• Gall, Frhr v. (1888), "Das vollständige Formensystem der binären Form 7terOrdnung", Mathematische Annalen31 (3): 318–336, doi:10.1007/BF01206218, ISSN 0025-5831, MR1510486

• Gordan, Paul (1868), "Beweis, dass jede Covariante und Invariante einer binären Form eine ganze Funktion mitnumerischen Coeffizienten einer endlichen Anzahl solcher Formen ist", J. F. Math 69 (69): 323–354,doi:10.1515/crll.1868.69.323

• Hilbert, David (1993) [1897], Theory of algebraic invariants [1], Cambridge University Press,ISBN 978-0-521-44457-6, MR1266168

• Kung, Joseph P. S.; Rota, Gian-Carlo (1984), "The invariant theory of binary forms" [1], American MathematicalSociety. Bulletin. New Series 10 (1): 27–85, doi:10.1090/S0273-0979-1984-15188-7, ISSN 0002-9904,MR722856

• Schur, Issai (1968), Grunsky, Helmut, ed., Vorlesungen über Invariantentheorie, Die Grundlehren dermathematischen Wissenschaften, 143, Berlin, New York: Springer-Verlag, ISBN 978-3540041399, MR0229674

• Shioda, Tetsuji (1967), "On the graded ring of invariants of binary octavics", American Journal of Mathematics89 (4): 1022–1046, doi:10.2307/2373415, ISSN 0002-9327, JSTOR 2373415, MR0220738

• Sturmfels, Bernd (1993), Algorithms in invariant theory, Texts and Monographs in Symbolic Computation,Berlin, New York: Springer-Verlag, doi:10.1007/978-3-211-77417-5, ISBN 978-3-211-82445-0, MR1255980

• Sylvester, J. J.; Franklin, F. (1879), "Tables of the Generating Functions and Groundforms for the BinaryQuantics of the First Ten Orders", American Journal of Mathematics 2 (3): 223–251, doi:10.2307/2369240,ISSN 0002-9327, MR1505222

Invariant of a binary form 41

References[1] http:/ / books. google. com/ books?isbn=0521449030

Invariant polynomialIn mathematics, an invariant polynomial is a polynomial that is invariant under a group acting on a vectorspace . Therefore is a -invariant polynomial if

for all and .Cases of particular importance are for Γ a finite group (in the theory of Molien series, in particular), a compactgroup, a Lie group or algebraic group. For a basis-independent definition of 'polynomial' nothing is lost by referringto the symmetric powers of the given linear representation of Γ.

References• This article incorporates material from Invariant polynomial on PlanetMath, which is licensed under the Creative

Commons Attribution/Share-Alike License.

Invariants of tensorsIn mathematics, in the fields of multilinear algebra and representation theory, invariants of tensors are coefficientsof the characteristic polynomial of the tensor A:

,where is the identity tensor and is the polynomials indeterminate (it is important to bear in mind that apolynomial's indeterminate may also be a non-scalar as long as power, scaling and adding make sense for it, e.g.,

is legitimate, and in fact, quite useful).The first invariant of an n×n tensor A ( ) is the coefficient for (coefficient for is always 1), the secondinvariant ( ) is the coefficient for , etc., the nth invariant is the free term.The definition of the invariants of tensors and specific notations used throughout the article were introduced into thefield of Rheology by Ronald Rivlin and became extremely popular there. In fact even the trace of a tensor isusually denoted as in the textbooks on rheology.

PropertiesThe first invariant (trace) is always the sum of the diagonal components:

The nth invariant is just , the determinant of (up to sign).The invariants do not change with rotation of the coordinate system (they are objective). Obviously, any function ofthe invariants only is also objective.

Invariants of tensors 42

Calculation of the invariants of symmetric 3×3 tensorsMost tensors used in engineering are symmetric 3×3. For this case the invariants can be calculated as:

(the sum of principal minors)

where , , are the eigenvalues of tensor A.Because of the Cayley–Hamilton theorem the following equation is always true:

where E is the second-order identity tensor.A similar equation holds for tensors of higher order.

Engineering applicationA scalar valued tensor function f that depends merely on the three invariants of a symmetric 3×3 tensor isobjective, i.e., independent from rotations of the coordinate system. Moreover, every objective tensor functiondepends only on the tensor's invariants. Thus, objectivity of a tensor function is fulfilled if, and only if, for somefunction we have

A common application to this is the evaluation of the potential energy as function of the strain tensor, within theframework of linear elasticity. Exhausting the above theorem the free energy of the system reduces to a function of 3scalar parameters rather than 6. Within linear elasticity the free energy has to be quadratic in the tensor's elements,which eliminates an additional scalar. Thus, for an isotropic material only two independent parameters are needed todescribe the elastic properties, known as the Lame coefficients. Consequently, experimental fits and computationalefforts may be eased significantly.

References

Kostant polynomial 43

Kostant polynomialIn mathematics, the Kostant polynomials, named after Bertram Kostant, provide an explicit basis of the ring ofpolynomials over the ring of polynomials invariant under the finite reflection group of a root system.

BackgroundIf the reflection group W corresponds to the Weyl group of a compact semisimple group K with maximal torus T,then the Kostant polynomials describe the structure of the de Rham cohomology of the generalized flag manifoldK/T, also isomorphic to G/B where G is the complexification of K and B is the corresponding Borel subgroup.Armand Borel showed that its cohomology ring is isomorphic to the quotient of the ring of polynomials by the idealgenerated by the invariant homogeneous polynomials of positive degree. This ring had already been considered byClaude Chevalley in establishing the foundations of the cohomology of compact Lie groups and their homogeneousspaces with André Weil, Jean-Louis Koszul and Henri Cartan; the existence of such a basis was used by Chevalleyto prove that the ring of invariants was itself a polynomial ring. A detailed account of Kostant polynomials wasgiven by Bernstein, Gelfand & Gelfand (1973) and independently Demazure (1973) as a tool to understand theSchubert calculus of the flag manifold. The Kostant polynomials are related to the Schubert polynomials definedcombinatorially by Lascoux & Schützenberger (1982) for the classical flag manifold, when G = SL(n,C). Theirstructure is governed by difference operators associated to the corresponding root system.Steinberg (1975) defined an analogous basis when the polynomial ring is replaced by the ring of exponentials of theweight lattice. If K is simply connected, this ring can be identified with the representation ring R(T) and theW-invariant subring with R(K). Steinberg's basis was again motivated by a problem on the topology of homogeneousspaces; the basis arises in describing the T-equivariant K-theory of K/T.

DefinitionLet Φ be a root system in a finite-dimensional real inner product space V with Weyl group W. Let Φ+ be a set ofpositive roots and Δ the corresponding set of simple roots. If α is a root, then sα denotes the corresponding reflectionoperator. Roots are regarded as linear polynomials on V using the inner product α(v) = (α,v). The choice of Δ givesrise to a Bruhat order on the Weyl group determined by the ways of writing elements minimally as products ofsimple root reflection. The minimal length for an elenet s is denoted . Pick an element v in V such that α(v) > 0for every positive root.If αi is a simple root with reflection operator si

then the corresponding divided difference operator is defined by

If and s has reduced expression

then

is independent of the reduced expression. Moreover

if and 0 otherwise.

Kostant polynomial 44

If w0 is the Coxeter element of W, the element of greatest length or equivalently the element sending Φ+ to −Φ+, then

More generally

for some constants as,t.Set

and

Then Ps is a homogeneous polynomial of degree .These polynomials are the Kostant polynomials.

PropertiesTheorem. The Kostant polynomials form a free basis of the ring of polynomials over the W-invariant polynomials.

In fact the matrix

is unitriangular for any total order such that s ≥ t implies .Hence

Thus if

with as invariant under W, then

Thus

where

another unitriangular matrix with polynomial entries. It can be checked directly that as is invariant under W.In fact δi satisfies the derivation property

Hence

Since

or 0, it follows that

Kostant polynomial 45

so that by the invertibility of N

for all i, i.e. at is invariant under W.

Steinberg basisAs above let Φ be a root system in a real inner product space V, and Φ+ a subset of positive roots. From these datawe obtain the subset Δ = { α1, α2, ..., αn} of the simple roots, the coroots

and the fundamental weights λ1, λ2, ..., λn as the dual basis of the coroots.For each element s in W, let Δs be the subset of Δ consisting of the simple roots satisfying s−1α < 0, and put

where the sum is calculated in the weight lattice P.The set of linear combinations of the exponentials eμ with integer coefficients for μ in P becomes a ring over Zisomorphic to the group algebra of P, or equivalently to the representation ring R(T) of T, where T is a maximal torusin K, the simply connected, connected compact semisimple Lie group with root system Φ. If W is the Weyl group ofΦ, then the representation ring R(K) of K can be identified with R(T)W.Steinberg's theorem. The exponentials λs (s in W) form a free basis for the ring of exponentials over the subring ofW-invariant exponentials.

Let ρ denote the half sum of the positive roots, and A denote the antisymmetrisation operator

The positive roots β with sβ positive can be seen as a set of positive roots for a root system on a subspace of V; theroots are the ones orthogonal to s.λs. The corresponding Weyl group equals the stabilizer of λs in W. It is generatedby the simple reflections sj for which sαj is a positive root.Let M and N be the matrices

where ψs is given by the weight s−1ρ - λs. Then the matrix

is triangular with respect to any total order on W such that s ≥ t implies . Steinberg proved that theentries of B are W-invariant exponential sums. Moreover its diagonal entries all equal 1, so it has determinant 1.Hence its inverse C has the same form. Define

If χ is an arbitrary exponential sum, then it follows that

with as the W-invariant exponential sum

Indeed this is the unique solution of the system of equations

Kostant polynomial 46

References• Bernstein, I. N.; Gelfand, I. M.; Gelfand, S. I. (1973), "Schubert cells, and the cohomology of the spaces G/P",

Russian Math. Surveys 28: 1–26, doi:10.1070/RM1973v028n03ABEH001557• Billey, Sara C. (1999), "Kostant polynomials and the cohomology ring for G/B.", Duke Math. J. 96: 205–224,

doi:10.1215/S0012-7094-99-09606-0• Bourbaki, Nicolas (1981), Groupes et algèbres de Lie, Chapitres 4, 5 et 6, Masson, ISBN 2-225-76076-4• Cartan, Henri (1950), "Notions d'algèbre différentielle; application aux groupes de Lie et aux variétés où opère un

groupe de Lie", Colloque de topologie (espaces fibrés), Bruxelles: 15–27• Cartan, Henri (1950), "La transgression dans un groupe de Lie et dans un espace fibré principal", Colloque de

topologie (espaces fibrés), Bruxelles: 57–71• Chevalley, Claude (1955), "Invariants of finite groups generated by reflections", Amer. J. Math. (The Johns

Hopkins University Press) 77 (4): 778–782, doi:10.2307/2372597, JSTOR 2372597• Demazure, Michel (1973), "Invariants symétriques entiers des groupes de Weyl et torsion", Invent. Math. 21:

287–301, doi:10.1007/BF01418790• Greub, Werner; Halperin, Stephen; Vanstone, Ray (1976), Connections, curvature, and cohomology. Volume III:

Cohomology of principal bundles and homogeneous spaces, Pure and Applied Mathematics, 47-III, AcademicPress

• Humphreys, James E. (1994), Introduction to Lie Algebras and Representation Theory (2nd ed.), Springer,ISBN 0387900535

• Kostant, Bertram (1963), "Lie algebra cohomology and generalized Schubert cells", Ann. Of Math. (Annals ofMathematics) 77 (1): 72–144, doi:10.2307/1970202, JSTOR 1970202

• Kostant, Bertram (1963), "Lie group representations on polynomial rings", Amer. J. Math. (The Johns HopkinsUniversity Press) 85 (3): 327–404, doi:10.2307/2373130, JSTOR 2373130

• Kostant, Bertram; Kumar, Shrawan (1986), "The nil Hecke ring and cohomology of G/P for a Kac–Moody groupG.", Proc. Nat. Acad. Sci. U.S.A. 83: 1543–1545, doi:10.1073/pnas.83.6.1543

• Alain, Lascoux; Schützenberger, Marcel-Paul (1982), "Polynômes de Schubert [Schubert polynomials]", C. R.Acad. Sci. Paris Sér. I Math. 294: 447–450

• McLeod, John (1979), The Kunneth formula in equivariant K-theory, Lecture Notes in Math., 741, Springer,pp. 316–333

• Steinberg, Robert (1975), "On a theorem of Pittie", Topology 14: 173–177, doi:10.1016/0040-9383(75)90025-7

LittlewoodRichardson rule 47

Littlewood–Richardson ruleIn mathematics, the Littlewood–Richardson rule is a combinatorial description of the coefficients that arise whendecomposing a product of two Schur functions as a linear combination of other Schur functions. These coefficientsare natural numbers, which the Littlewood–Richardson rule describes as counting certain skew tableaux. They occurin many other mathematical contexts, for instance as multiplicity in the decomposition of tensor products ofirreducible representations of general linear groups (or related groups like the special linear and special unitarygroups), or in the decomposition of certain induced representations in the representation theory of the symmetricgroup, or in the area of algebraic combinatorics dealing with Young tableaux and symmetric polynomials

Littlewood–Richardson coefficients depend on three partitions, say , of which and describe the Schurfunctions being multiplied, and gives the Schur function of which this is the coefficient in the linear combination;in other words they are the coefficients such that

The Littlewood–Richardson rule states that is equal to the number of Littlewood–Richardson tableaux of shapeand of weight .

HistoryUnfortunately the Littlewood–Richardson rule is much harder to prove than was at first suspected. The author was once told that theLittlewood–Richardson rule helped to get men on the moon but was not proved until after they got there.

Gordon James (1987)

The Littlewood–Richardson rule was first stated by D. E. Littlewood and A. R. Richardson (1934, theorem III p.119) but though they claimed it as a theorem they only proved it in some fairly simple special cases.Robinson (1938) claimed to complete their proof, but his argument had gaps, though it was so obscurely written thatthese gaps were not noticed for some time, and his argument is reproduced in the book (Littlewood 1950). Some ofthe gaps were later filled by Macdonald (1995). The first rigorous proofs of the rule were given four decades after itwas found, by Schützenberger (1977) and Thomas (1974), after the necessary combinatorial theory was developedby C. Schensted (1961), Schützenberger (1963), and Knuth (1970) in their work on the Robinson–Schenstedcorrespondence. There are now several short proofs of the rule, such as (Gasharov 1998), and (Stembridge 2002)using Bender-Knuth involutions. Littelmann (1994) used the Littelmann path model to generalize theLittlewood–Richardson rule to other semisimple Lie groups.The Littlewood–Richardson rule is notorious for the number of errors that appeared prior to its complete, publishedproof. Several published attempts to prove it are incomplete, and it is particularly difficult to avoid errors whendoing hand calculations with it: even the original example in D. E. Littlewood and A. R. Richardson (1934) containsan error.

LittlewoodRichardson rule 48

Littlewood–Richardson ruleThe Littlewood–Richardson rule states that is equal to the number of Littlewood–Richardson tableaux of shape

and of weight .

Littlewood–Richardson tableaux

A Littlewood–Richardson tableau

A Littlewood–Richardson tableau is a skew semistandard tableau withthe additional property that the sequence obtained by concatenating itsreversed rows is a lattice word (or lattice permutation), which meansthat in every initial part of the sequence any number occurs at leastas often as the number . Another equivalent (though not quiteobviously so) characterization is that the tableau itself, and any tableauobtained from it by removing some number of its leftmost columns,has a weakly decreasing weight. Many other combinatorial notionshave been found that turn out to be in bijection withLittlewood–Richardson tableaux, and can therefore also be used to define the Littlewood–Richardson coefficients.

Another Littlewood–Richardson tableau

Example

Consider the case that , and . Then the fact that can bededuced from the fact that the two tableaux shown at the right are the only two Littlewood–Richardson tableaux ofshape and weight . Indeed, since the last box on the first nonempty line of the skew diagram can onlycontain an entry 1, the entire first line must be filled with entries 1 (this is true for any Littlewood–Richardsontableau); in the last box of the second row we can only place a 2 by column strictness and the fact that our latticeword cannot contain any larger entry before it contains a 2. For the first box of the second row we can now either usea 1 or a 2. Once that entry is chosen, the third row must contain the remaining entries to make the weight (3,2,1), in aweakly increasing order, so we have no choice left any more; in both case it turns out that we do find aLittlewood–Richardson tableau.

A more geometrical descriptionThe condition that the sequence of entries read from the tableau in a somewhat peculiar order form a lattice word, can be replaced by a more local and geometrical condition. Since in a semistandard tableau equal entries never occur in the same column, one can number the copies of any value from right to left, which is their order in of occurrence in the sequence that should be a lattice word. Call the number so associated to each entry its index, and write an entry i with index j as i[j]. Now if some Littlewood–Richardson tableau contains an entry occurs with index j, then that entry i[j] should occur in a row strictly below that of (which certainly also occurs, since the entry i − 1 occurs as least as often as the entry i does). In fact the entry i[j] should also occur in a column no further to the right than that same entry (which at first sight appears to be a stricter condition). If the weight of the Littlewood–Richardson tableau is fixed beforehand, then one can form a fixed collection of indexed entries, and the

LittlewoodRichardson rule 49

if these are placed in a way respecting those geometric restrictions, in addition to those of semistandard tableaux andthe condition that indexed copies of the same entries should respect right-to-left ordering of the indexes, then theresulting tableaux are guaranteed to be Littlewood–Richardson tableaux.

An algorithmic form of the ruleThe Littlewood–Richardson as stated above gives a combinatorial expression for individual Littlewood–Richardsoncoefficients, but gives no indication of a practical method to enumerate the Littlewood–Richardson tableaux in orderto find the values of these coefficients. Indeed for given there is no simple criterion to determine whetherany Littlewood–Richardson tableaux of shape and of weight exist at all (although there are a number ofnecessary conditions, the simplest of which is ); therefore it seems inevitable that in some casesone has to go through an elaborate search, only to find that no solutions exist.Nevertheless, the rule leads to a quite efficient procedure to determine the full decomposition of a product of Schurfunctions, in other words to determine all coefficients for fixed λ and μ, but varying ν. This fixes the weight ofthe Littlewood–Richardson tableaux to be constructed and the "inner part" λ of their shape, but leaves the "outerpart" ν free. Since the weight is known, the set of indexed entries in the geometric description is fixed. Now forsuccessive indexed entries, all possible positions allowed by the geometric restrictions can be tried in a backtrackingsearch. The entries can be tried in increasing order, while among equal entries they can be tried by decreasing index.The latter point is the key to efficiency of the search procedure: the entry i[j] is then restricted to be in a column tothe right of , but no further to the right than (if such entries are present). This strongly restrictsthe set of possible positions, but always leaves at least one valid position for ; thus every placement of an entrywill give rise to at least one complete Littlewood–Richardson tableau, and the search tree contains no dead ends.A similar method can be used to find all coefficients for fixed λ and ν, but varying μ.

Littlewood–Richardson coefficientsThe Littlewood–Richardson coefficients c   appear in the following ways:• They are the structure constants for the product in the ring of symmetric functions with respect to the basis of

Schur functions

or equivalently c   is the inner product of sν and sλsμ.• They express skew Schur functions in terms of Schur functions

• The c   appear as intersection numbers on a Grassmannian:

where σμ is the class of the Schubert variety of a Grassmannian corresponding to μ.• c   is the number of times the irreducible representation Vλ ⊗ Vμ of the product of symmetric groups S|λ| × S|μ|

appears in the restriction of the representation Vν of S|ν| to S|λ| × S|μ|. By Frobenius reciprocity this is also thenumber of times that Vν occurs in the representation of S|ν| induced from Vλ ⊗ Vμ.

• The c   appear in the decomposition of the tensor product (Fulton 1997) of two Schur modules (irreduciblerepresentations of special linear groups)

• c   is the number of standard Young tableaux of shape ν/μ that are jeu de taquin equivalent to some fixed standardYoung tableau of shape λ.

LittlewoodRichardson rule 50

• c   is the number of Littlewood–Richardson tableaux of shape ν/λ and of weight μ.• c   is the number of pictures between μ and ν/λ.

Generalizations and special casesZelevinsky (1981) extended the Littlewood–Richardson rule to skew Schur functions as follows:

where the sum is over all tableaux T on μ/ν such that for all j, the sequence of integers λ+ω(T≥j) is non-increasing,and ω is the weight.Pieri's formula, which is the special case of the Littlewood–Richardson rule in the case when one of the partitionshas only one part, states that

•

where Sn is the Schur function of a partition with one row and the sum is over all partitions λ obtained from μ byadding n elements to its Ferrers diagram, no two in the same column.If both partitions are rectangular in shape, the sum is also multiplicity free (Okada 1998). Fix a, b, p, and q positiveintegers with p q. Denote by the partition with p parts of length a. The partitions indexing nontrivialcomponents of are those partitions with length such that•••For example,

.

LittlewoodRichardson rule 51

ExamplesThe examples of Littlewood-Richardson coefficients below are given in terms of products of Schur polynomials Sπ,indexed by partitions π, using the formula

All coefficients with ν at most 4 are given by:• S0Sπ = Sπ for any π. where S0=1 is the Schur polynomial of the empty partition• S1S1 = S2 + S11• S2S1 = S3 + S21• S11S1 = S111 + S21• S3S1 = S4 + S31• S21S1 = S31 + S22 + S211• S2S2 = S4 + S31 + S22• S2S11 = S31 + S211• S111S1 = S1111 + S211• S11S11 = S1111 + S211 + S22Most of the coefficients for small partitions are 0 or 1, which happens in particular whenever one of the factors is ofthe form Sn or S11...1, because of Pieri's formula and its transposed counterpart. The simplest example with acoefficient larger than 1 happens when neither of the factors has this form:• S21S21 = S42 + S411 + S33 + 2S321 + S3111 + S222 + S2211.For larger partitions the coefficients become more complicated. For example,• S321S321 = S642 +S6411 +S633 +2S6321 +S63111 +S6222 +S62211 +S552 +S5511 +2S543 +4S5421 +2S54111 +3S5331

+3S5322 +4S53211 +S531111 +2S52221 +S522111 +S444 +3S4431 +2S4422 +3S44211 +S441111 +3S4332 +3S43311 +4S43221+2S432111 +S42222 +S422211 +S3333 +2S33321 +S333111 +S33222 +S332211 with 34 terms and total multiplicity 62, andthe largest coefficient is 4

• S4321S4321 is a sum of 206 terms with total multiplicity is 930, and the largest coefficient is 18.• S54321S54321 is a sum of 1433 terms with total multiplicity 26704, and the largest coefficient (that of S86543211) is

176.• S654321S654321 is a sum of 10873 terms with total multiplicity is 1458444 (so the average value of the coefficients

is more than 100, and they can be as large as 2064).The original example given by Littlewood & Richardson (1934, p. 122-124) was (after correcting for 3 tableaux theyfound but forgot to include in the final sum)• S431S221 = S652 + S6511 + S643 + 2S6421 + S64111 + S6331 + S6322 + S63211 + S553 + 2S5521 + S55111 + 2S5431 +

2S5422 + 3S54211 + S541111 + S5332 + S53311 + 2S53221 + S532111 + S4432 + S44311 + 2S44221 + S442111 + S43321 +S43222 + S432211

with 26 terms coming from the following 34 tableaux:

....11 ....11 ....11 ....11 ....11 ....11 ....11 ....11 ....11

...22 ...22 ...2 ...2 ...2 ...2 ... ... ...

.3 . .23 .2 .3 . .22 .2 .2

3 3 2 2 3 23 2

3 3

....1 ....1 ....1 ....1 ....1 ....1 ....1 ....1 ....1

...12 ...12 ...12 ...12 ...1 ...1 ...1 ...2 ...1

.23 .2 .3 . .23 .22 .2 .1 .2

LittlewoodRichardson rule 52

3 2 2 2 3 23 23 2

3 3

....1 ....1 ....1 ....1 ....1 ....1 ....1 ....1

...2 ...2 ...2 ... ... ... ... ...

.1 .3 . .12 .12 .1 .2 .2

2 1 1 23 2 22 13 1

3 2 2 3 3 2 2

3 3

.... .... .... .... .... .... .... ....

...1 ...1 ...1 ...1 ...1 ... ... ...

.12 .12 .1 .2 .2 .11 .1 .1

23 2 22 13 1 22 12 12

3 3 2 2 3 23 2

3 3

Calculating skew Schur functions is similar. For example, the 15 Littlewood–Richardson tableaux for ν=5432 andλ=331 are

...11 ...11 ...11 ...11 ...11 ...11 ...11 ...11 ...11 ...11 ...11 ...11 ...11 ...11 ...11

...2 ...2 ...2 ...2 ...2 ...2 ...2 ...2 ...2 ...2 ...2 ...2 ...2 ...2 ...2

.11 .11 .11 .12 .11 .12 .13 .13 .23 .13 .13 .12 .12 .23 .23

12 13 22 12 23 13 12 24 14 14 22 23 33 13 34

so S5432/331 = Σc  Sμ = S52 + S511 + S4111 + S2221 + 2S43 + 2S3211 + 2S322 + 2S331 + 3S421 (Fulton 1997, p. 64).

References• Fulton, William (1997), Young tableaux, London Mathematical Society Student Texts, 35, Cambridge University

Press, p. 121, ISBN 978-0-521-56144-0; 978-0-521-56724-4, MR1464693• Gasharov, Vesselin (1998), "A short proof of the Littlewood-Richardson rule", European Journal of

Combinatorics 19 (4): 451–453, doi:10.1006/eujc.1998.0212, ISSN 0195-6698, MR1630540• James, Gordon (1987), "The representation theory of the symmetric groups", The Arcata Conference on

Representations of Finite Groups (Arcata, Calif., 1986), Proc. Sympos. Pure Math., 47, Providence, R.I.:American Mathematical Society, pp. 111–126, MR933355

• Knuth, Donald E. (1970), "Permutations, matrices, and generalized Young tableaux" [1], Pacific Journal ofMathematics 34: 709–727, ISSN 0030-8730, MR0272654

• Littelmann, Peter (1994), "A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras", Invent. Math.116: 329–346, doi:10.1007/BF01231564

• Littlewood, Dudley E. (1950), The theory of group characters and matrix representations of groups [2], AMSChelsea Publishing, Providence, RI, ISBN 978-0-8218-4067-2, MR00002127

• Littlewood, D. E.; Richardson, A. R. (1934), "Group Characters and Algebra" [3], Philosophical Transactions ofthe Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character (The RoyalSociety) 233 (721–730): 99–141, doi:10.1098/rsta.1934.0015, ISSN 0264-3952

• Macdonald, I. G. (1995), Symmetric functions and Hall polynomials [1], Oxford Mathematical Monographs (2nded.), The Clarendon Press Oxford University Press, ISBN 978-0-19-853489-1, MR1354144

• Okada, Soichi (1998), "Applications of minor summation formulas to rectangular-shaped representations of classical groups", Journal of Algebra 205 (2): 337–367, doi:10.1006/jabr.1997.7408, ISSN 0021-8693,

LittlewoodRichardson rule 53

MR1632816• Robinson, G. de B. (1938), "On the Representations of the Symmetric Group", American Journal of Mathematics

(The Johns Hopkins University Press) 60 (3): 745–760, doi:10.2307/2371609, ISSN 0002-9327, JSTOR 2371609Zbl0019.25102 [4]

• Schensted, C. (1961), "Longest increasing and decreasing subsequences" [5], Canadian Journal of Mathematics13: 179–191, doi:10.4153/CJM-1961-015-3, ISSN 0008-414X, MR0121305

• Schützenberger, M. P. (1963), "Quelques remarques sur une construction de Schensted" [6], MathematicaScandinavica 12: 117–128, ISSN 0025-5521, MR0190017

• Schützenberger, Marcel-Paul (1977), "La correspondance de Robinson", Combinatoire et représentation dugroupe symétrique (Actes Table Ronde CNRS, Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976), Lecture notesin mathematics, 579, Berlin, New York: Springer-Verlag, pp. 59–113, doi:10.1007/BFb0090012,ISBN 978-3-540-08143-2, MR0498826

• Stembridge, John R. (2002), "A concise proof of the Littlewood-Richardson rule" [7], Electronic Journal ofCombinatorics 9 (1): Note 5, 4 pp. (electronic), ISSN 1077-8926, MR1912814

• Thomas, Glânffrwd P. (1974), Baxter algebras and Schur functions, Ph.D. Thesis, Swansea: University Collegeof Swansea

• van Leeuwen, Marc A. A. (2001), "The Littlewood-Richardson rule, and related combinatorics" [8], Interaction ofcombinatorics and representation theory, MSJ Mem., 11, Tokyo: Math. Soc. Japan, pp. 95–145, MR1862150

• Zelevinsky, A. V. (1981), "A generalization of the Littlewood-Richardson rule and theRobinson-Schensted-Knuth correspondence", Journal of Algebra 69 (1): 82–94,doi:10.1016/0021-8693(81)90128-9, ISSN 0021-8693, MR613858

External links• An online program [9], decomposing products of Schur functions using the Littlewood–Richardson rule

References[1] http:/ / projecteuclid. org/ euclid. pjm/ 1102971948[2] http:/ / www. ams. org/ bookstore?fn=20& arg1=alggeom& item=CHEL-357-H[3] http:/ / www. jstor. org/ stable/ 91293[4] http:/ / www. zentralblatt-math. org/ zmath/ en/ search/ ?q=an:0019. 25102[5] http:/ / books. google. com/ ?id=G3sZ2zG8AiMC[6] http:/ / gdz. sub. uni-goettingen. de/ no_cache/ dms/ load/ img/ ?IDDOC=221996[7] http:/ / www. emis. de/ journals/ EJC/ Volume_9/ PDF/ v9i1n5. pdf[8] http:/ / www-math. univ-poitiers. fr/ ~maavl/ pdf/ lrr. pdf[9] http:/ / young. sp2mi. univ-poitiers. fr/ cgi-bin/ form-prep/ marc/ LiE_form. act?action=LRR

Modular invariant of a group 54

Modular invariant of a groupIn mathematics, a modular invariant of a group is an invariant of a finite group acting on a vector space of positivecharacteristic (usually dividing the order of the group). The study of modular invariants was originated in about 1914by Dickson (2004).

Dickson invariantWhen G is the finite general linear group GLn(Fq) over the finite field Fq of order a prime power q acting on the ringFq[X1, ...,Xn] in the natural way, Dickson (1911) found a complete set of invariants as follows. Write [e1, ...,en] forthe determinant of the matrix whose entries are X , where e1, ...,en are

non-negative integers. For example, the Moore determinant [0,1,2] of order 3 is

Then under the action of an element g of GLn(Fq) these determinants are all multiplied by det(g), so they are allinvariants of SLn(Fp) and the ratio [e1, ...,en]/[0,1,...,n−1] are invariants of GLn(Fq), called Dickson invariants.Dickson proved that the full ring of invariants Fq[X1, ...,Xn]GLn(Fq) is a polynomial algebra over the n Dicksoninvariants [0,1,...,i−1,i+1,...,n]/[0,1,...,n−1] for i=0, 1, ..., n−1. Steinberg (1987) gave a shorter proof of Dickson'stheorem.The matrices [e1, ...,en] are divisible by all non-zero linear forms in the variables Xi with coefficients in the finitefield Fq. In particular the Moore determinant [0,1,...,n−1] is a product of such linear forms, taken over1+q+q2+...+qn–1 representatives of n–1 dimensional projective space over the field. This factorization is similar tothe factorization of the Vandermonde determinant into linear factors.

References• Dickson, Leonard Eugene (1911), "A Fundamental System of Invariants of the General Modular Linear Group

with a Solution of the Form Problem", Transactions of the American Mathematical Society (Providence, R.I.:American Mathematical Society) 12 (1): 75–98, ISSN 0002-9947, JSTOR 1988736

• Dickson, Leonard Eugene (2004) [1914], On invariants and the theory of numbers [1], Dover Phoenix editions,New York: Dover Publications, ISBN 978-0-486-43828-3, MR0201389

• Rutherford, Daniel Edwin (2007) [1932], Modular invariants [2], Cambridge Tracts in Mathematics andMathematical Physics, No. 27, Ramsay Press, ISBN 978-1-4067-3850-6, MR0186665

• Sanderson, Mildred (1913), "Formal Modular Invariants with Application to Binary Modular Covariants",Transactions of the American Mathematical Society (Providence, R.I.: American Mathematical Society) 14 (4):489–500, ISSN 0002-9947, JSTOR 1988702

• Steinberg, Robert (1987), "On Dickson's theorem on invariants" [3], Journal of the Faculty of Science. Universityof Tokyo. Section IA. Mathematics 34 (3): 699–707, ISSN 0040-8980, MR927606

Modular invariant of a group 55

References[1] http:/ / books. google. com/ books?isbn=0486438287[2] http:/ / www. archive. org/ details/ modularinvariant033204mbp[3] http:/ / repository. dl. itc. u-tokyo. ac. jp/ dspace/ bitstream/ 2261/ 1682/ 1/ jfs340309. pdf

Moduli spaceIn algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose pointsrepresent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spacesfrequently arise as solutions to classification problems: If one can show that a collection of interesting objects (e.g.,the smooth algebraic curves of a fixed genus) can be given the structure of a geometric space, then one canparametrize such objects by introducing coordinates on the resulting space. In this context, the term "modulus" isused synonymously with "parameter"; moduli spaces were first understood as spaces of parameters rather than asspaces of objects.

Basic Examples

Projective Space and GrassmaniansThe real projective space Pn is a moduli space. It is the space of lines in Rn+1 which pass through the origin.Similarly, complex projective space is the space of all complex lines in Cn+1 passing through the origin.More generally, the Grassmannian G(k, V) of a vector space V over a field F is the moduli space of all k-dimensionallinear subspaces of V.

Hilbert SchemeThe Hilbert scheme Hilb(X) is a moduli scheme. Every closed point of Hilb(X) corresponds to a closed subschemeof a fixed scheme X, and every closed subscheme is represented by such a point.

DefinitionsThere are several different related notions of what it means for a space M to be a moduli space. Each of thesedefinitions formalizes a different notion of what it means for the points of a space to represent geometric objects.

Fine Moduli SpacesThis is the most important notion. Heuristically, if we have a space M for which each point m∈ M corresponds to analgebro-geometric object Um, then we can assemble these objects into a tautological family U over M. (For example,the Grassmanian G(k, V) carries a rank k bundle whose fiber at any point [L] ∈ G(k, V) is simply the linear subspaceL ⊂ V.) We say that such a family is universal if any family of algebro-geometric objects T over any base space B isthe pullback of U along a unique map B → M. A fine moduli space is a space M which is the base of a universalfamily.More precisely, suppose that we have a functor F from schemes to sets, which assigns to a scheme B the set of allsuitable families of objects with base B. A space M is a fine moduli space for the functor F if M represents F, i.e.,the functor of points Hom(−,M) is naturally isomorphic to F. This implies that M carries a universal family; thisfamily is the family on M corresponding to the identity map 1M ∈ Hom(M, M).

Moduli space 56

Coarse Moduli SpacesFine moduli spaces are very useful, but they do not always exist and are frequently difficult to construct, somathematicians sometimes use a weaker notion, the idea of a coarse moduli space. A space M is a coarse modulispace for the functor F if there exists a natural transformation τ: F → Hom(−,M) and τ is universal among suchnatural transformations. More concretely, M is a coarse moduli space for F if any family T over a base B gives rise toa map φT: B → M and any two objects V and W (regarded as families over a point) correspond to the same point ofM if and only if V and W are isomorphic. Thus, M is a space which has a point for every object that could appear in afamily, and whose geometry reflects the ways objects can vary in families. Note, however, that a coarse modulispace does not necessarily carry any family of appropriate objects, let alone a universal one.In other words, a fine moduli space includes both a base space M and universal family T → M, while a coarse modulispace only has the base space M.

Moduli stacksIt is frequently the case that interesting geometric objects come equipped with lots of natural automorphisms. This inparticular makes the existence of a fine moduli space impossible (intuitively, the idea is that if L is some geometricobject, the trivial family L × [0,1] can be made into a twisted family on the circle S1 by identifying L × {0} with L ×{1} via a nontrivial automorphism. Now if a fine moduli space X existed, the map S1 → X should not be constant,but would have to be constant on any proper open set by triviality), one can still sometimes obtain a coarse modulispace. However, this approach is not ideal, as such spaces are not guaranteed to exist, are frequently singular whenthey do exist, and miss details about some non-trivial families of objects they classify.A more sophisticated approach is to enrich the classification by remembering the isomorphisms. More precisely, onany base B one can consider the category of families on B with only isomorphisms between families taken asmorphisms. One then considers the fibred category which assigns to any space B the groupoid of families over B.The use of these categories fibred in groupoids to describe a moduli problem goes back to Grothendieck (1960/61).In general they cannot be represented by schemes or even algebraic spaces, but in many cases they have a naturalstructure of an algebraic stack.Algebraic stacks and their use to analyse moduli problems appeared in Deligne-Mumford (1969) as a tool to provethe irreducibility of the (coarse) moduli space of curves of a given genus. The language of algebraic stacksessentially provides a systematic way to view the fibred category that constitutes the moduli problem as a "space",and the moduli stack of many moduli problems is better-behaved (such as smooth) than the corresponding coarsemoduli space.

Further Examples

Moduli of curves

The moduli stack classifies families of smooth projective curves of genus , together with theirisomorphisms. When g > 1, this stack may be compactified by adding new "boundary" points which correspond tostable nodal curves (together with their isomorphisms). A curve is stable if it has only a finite group ofautomorphisms. The resulting stack is denoted . Both moduli stacks carry universal families of curves. One canalso define coarse moduli spaces representing isomorphism classes of smooth or stable curves. These coarse modulispaces were actually studied before the notion of moduli stack was invented. In fact, the idea of a moduli stack wasinvented by Deligne and Mumford in an attempt to prove the projectivity of the coarse moduli spaces. In recentyears, it has become apparent that the stack of curves is actually the more fundamental object.Both stacks above have dimension ; hence a stable nodal curve can be completely specified by choosing the values of 3g-3 parameters, when g > 1. In lower genus, one must account for the presence of smooth families of

Moduli space 57

automorphisms, by subtracting their number. There is exactly one complex curve of genus zero, the Riemann sphere,and its group of isomorphisms is PGL(2). Hence the dimension of is

dim(space of genus zero curves) - dim(group of automorphisms) = 0 - dim(PGL(2)) = -3.Likewise, in genus 1, there is a one-dimensional space of curves, but every such curve has a one-dimensional groupof automorphisms. Hence the stack has dimension 0. The coarse moduli spaces have the same dimension as thestacks when g > 1; however, in genus zero the coarse moduli space has dimension zero, and in genus one, it hasdimension one.One can also enrich the problem by considering the moduli stack of genus g nodal curves with n marked points. Suchmarked curves are said to be stable if the subgroup of curve automorphisms which fix the marked points is finite.The resulting moduli stacks of smooth (or stable) genus g curves with n-marked points are denoted (or

), and have dimension 3g-3 + n.

A case of particular interest is the moduli stack of genus 1 curves with one marked point. This is the stack ofelliptic curves, and is the natural home of the much studied modular forms, which are meromorphic sections ofbundles on this stack.

Moduli of varietiesIn higher dimensions, moduli of algebraic varieties are more difficult to construct and study. For instance, the higherdimensional analogue of the moduli space of elliptic curves discussed above is the moduli space of abelian varieties.This is the problem underlying Siegel modular form theory. See also Shimura variety.

Moduli of vector bundlesAnother important moduli problem is to understand the geometry of (various substacks of) the moduli stack

of rank n vector bundles on a fixed algebraic variety X. This stack has been most studied when X isone-dimensional, and especially when n equals one. In this case, the coarse moduli space is the Picard scheme,which like the moduli space of curves, was studied before stacks were invented. Finally, when the bundles have rank1 and degree zero, the study of the coarse moduli space is the study of the Jacobian variety.In applications to physics, the number of moduli of vector bundles and the closely related problem of the number ofmoduli of principal G-bundles has been found to be significant in gauge theory.

Methods for constructing moduli spacesThe modern formulation of moduli problems and definition of moduli spaces in terms of the moduli functors (ormore generally the categories fibred in groupoids), and spaces (almost) representing them, dates back toGrothendieck (1960/61), in which he described the general framework, approaches and main problems usingTeichmüller spaces in complex analytical geometry as an example. The talks in particular describe the generalmethod of constructing moduli spaces by first rigidifying the moduli problem under consideration.More precisely, the existence of non-trivial automorphisms of the objects being classified makes it impossible to have a fine moduli space. However, it is often possible to consider a modified moduli problem of classifying the original objects together with additional data, chosen in such a way that the identity is the only automorphism respecting also the additional data. With a suitable choice of the rigidifying data, the modified moduli problem will have a (fine) moduli space T, often described as a subscheme of a suitable Hilbert scheme or Quot scheme. The rigidifying data is moreover chosen so that it corresponds to a principal bundle with an algebraic structure group G. Thus one can move back from the rigidified problem to the original by taking quotient by the action of G, and the problem of constructing the moduli space becomes that of finding a scheme (or more general space) that is (in a suitably strong sense) the quotient T/G of T by the action of G. The last problem in general does not admit a solution; however, it is addressed by the groundbreaking geometric invariant theory (GIT), developed by David Mumford in

Moduli space 58

Mumford (1965), which shows that under suitable conditions the quotient indeed exists.To see how this might work, consider the problem of parametrizing smooth curves of genus g > 2. A smooth curvetogether with a complete linear system of degree d > 2g is equivalent to a closed one complex dimensionalsubscheme of the projective space Pd − g. Consequently, the moduli space of smooth curves and linear systems(satisfying certain criteria) may be embedded in the Hilbert scheme of a sufficiently high-dimensional projectivespace. This locus H in the Hilbert scheme has an action of PGL(n) which mixes the elements of the linear system;consequently, the moduli space of smooth curves is then recovered as the quotient of H by the projective generallinear group.Another general approach is primarily associated with Michael Artin. Here the idea is to start with any object of thekind to be classified and study its deformation theory. This means first constructing infinitesimal deformations, thenappealing to prorepresentability theorems to put these together into an object over a formal base. Next an appeal toGrothendieck's formal existence theorem provides an object of the desired kind over a base which is a complete localring. This object can be approximated via Artin's approximation theorem by an object defined over a finitelygenerated ring. The spectrum of this latter ring can then be viewed as giving a kind of coordinate chart on the desiredmoduli space. By gluing together enough of these charts, we can cover the space, but the map from our union ofspectra to the moduli space will in general be many to one. We therefore define an equivalence relation on theformer; essentially, two points are equivalent if the objects over each are isomorphic. This gives a scheme and anequivalence relation, which is enough to define an algebraic space (actually an algebraic stack if we are beingcareful) if not always a scheme.

In PhysicsThe term moduli space is sometimes used in physics to refer specifically the moduli space of vacuum expectationvalues of a set of scalar fields, or to the moduli space of possible string backgrounds.Moduli spaces also appear in physics in cohomological field theory, where one can use Feynman path integrals tocompute the intersection numbers of various algebraic moduli spaces.

References• Grothendieck, Alexander (1960/1961). "Techniques de construction en géométrie analytique. I. Description

axiomatique de l'espace de Teichmüller et de ses variantes." (http:/ / archive. numdam. org/ article/SHC_1960-1961__13_1_A4_0. pdf). Séminaire Henri Cartan 13 no. 1, Exposés No. 7 and 8 (Paris: SecrétariatMathématique)

• Mumford, David, Geometric invariant theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge,Band 34 Springer-Verlag, Berlin-New York 1965 vi+145 pp MR0214602

• Mumford, David; Fogarty, J.; Kirwan, F. Geometric invariant theory. Third edition. Ergebnisse der Mathematikund ihrer Grenzgebiete (2) (Results in Mathematics and Related Areas (2)), 34. Springer-Verlag, Berlin, 1994.xiv+292 pp. MR1304906 ISBN 3-540-56963-4

• Papadopoulos, Athanase, ed. (2007), Handbook of Teichmüller theory. Vol. I, IRMA Lectures in Mathematicsand Theoretical Physics, 11, European Mathematical Society (EMS), Zürich, doi:10.4171/029, ISBN978-3-03719-029-6, MR2284826

• Papadopoulos, Athanase, ed. (2009), Handbook of Teichmüller theory. Vol. II, IRMA Lectures in Mathematicsand Theoretical Physics, 13, European Mathematical Society (EMS), Zürich, doi:10.4171/055, ISBN978-3-03719-055-5, MR2524085

• Deligne, Pierre; Mumford, David (1969). "The irreducibility of the space of curves of given genus" (http:/ /archive. numdam. org/ article/ PMIHES_1969__36__75_0. pdf). Publications Mathématiques de l'IHÉS (Paris)36: 75–109.

Moduli space 59

• Harris, Joe; Morrison, Ian (1998). Moduli of Curves. Springer Verlag. ISBN 0387984291.• Katz, Nicholas M; Mazur, Barry (1985). Arithmetic Moduli of Elliptic Curves. Princeton University Press.

ISBN 0691083525.• Faltings, Gerd; Chai, Ching-Li (1990). Degeneration of Abelian Varieties. Springer Verlag. ISBN 3540520155.• Viehweg, Eckart (1995). Quasi-Projective Moduli for Polarized Manifolds (http:/ / www. uni-due. de/ ~mat903/

books/ vibuch. pdf). Springer Verlag. ISBN 3540592555.• Simpson, Carlos (1994). "Moduli of representations of the fundamental group of a smooth projective variety I"

(http:/ / archive. numdam. org/ article/ PMIHES_1994__79__47_0. pdf). Publications Mathématiques de l'IHÉS(Paris) 79: 47–129.

Molien seriesIn mathematics, a Molien series is a generating function attached to a linear representation ρ of a group G on afinite-dimensional vector space V. It counts the homogeneous polynomials of a given total degree d that areinvariants for G. It is named for Theodor Molien.

FormulationMore formally, there is a vector space of such polynomials, for each given value of d = 0, 1, 2, ..., and we write ndfor its vector space dimension, or in other words the number of linearly independent homogeneous invariants of agiven degree. In more algebraic terms, take the d-th symmetric power of V, and the representation of G on it arisingfrom ρ. The invariants form the subspace consisting of all vectors fixed by all elements of G, and nd is its dimension.The Molien series is then by definition the formal power series

This can be looked at another way, by considering the representation of G on the symmetric algebra of V, and thenthe whole subalgebra R of G-invariants. Then nd is the dimension of the homogeneous part of R of dimension d,when we look at it as graded ring. In this way a Molien series is also a kind of Hilbert function. Without furtherhypotheses not a great deal can be said, but assuming some conditions of finiteness it is then possible to show thatthe Molien series is a rational function. The case of finite groups is most often studied.

FormulaMolien showed that

This means that the coefficient of td in this series is the dimension nd defined above. It assumes that the characteristicof the field does not divide |G| (but even without this assumption, Molien's formula in the form

is valid, although it does not help with computing M(t)).

Molien series 60

ExampleConsider acting on R3 by permuting the coordinates. Note that is constant on conjugacy classes,so it is enough to take one from each of the three classes in ; so

and whereand .

Then

References• David A. Cox, John B. Little, Donal O'Shea (2005), Using Algebraic Geometry, pp. 295–8• Molien, Th. (1897). "Uber die Invarianten der linearen Substitutionsgruppen." [1]. Sitzungber. Konig. Preuss.

Akad. Wiss. (J. Berl. Ber.) 52: 1152–1156. JFM 28.0115.01.• Mukai, S. (2002). An introduction to invariants and moduli [2]. Cambridge Studies in Advanced Mathematics. 81.

ISBN 9780521809061.

References[1] http:/ / books. google. com/ ?id=EIxK-opAmJYC& pg=PA1152[2] http:/ / www. cambridge. org/ catalogue/ catalogue. asp?isbn=0521809061

Newton's identitiesIn mathematics, Newton's identities, also known as the Newton–Girard formulae, give relations between twotypes of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated atthe roots of a monic polynomial P in one variable, they allow expressing the sums of the k-th powers of all roots of P(counted with their multiplicity) in terms of the coefficients of P, without actually finding those roots. Theseidentities were found by Isaac Newton around 1666, apparently in ignorance of earlier work (1629) by Albert Girard.They have applications in many areas of mathematics, including Galois theory, invariant theory, group theory,combinatorics, as well as further applications outside mathematics, including general relativity.

Mathematical statement

Formulation in terms of symmetric polynomialsLet x1,…, xn be variables, denote for k ≥ 1 by pk(x1,…,xn) the k-th power sum:

and for k ≥ 0 denote by ek(x1,…,xn) the elementary symmetric polynomial that is the sum of all distinct products of kdistinct variables, so in particular

Then the Newton's identities can be stated as

Newton's identities 61

valid for all k ≥ 1. Concretely, one gets for the first few values of k:

The form and validity of these equations do not depend on the number n of variables (although the point where theleft-hand side becomes 0 does, namely after the n-th identity), which makes it possible to state them as identities inthe ring of symmetric functions. In that ring one has

and so on; here the left-hand sides never become zero. These equations allow to recursively express the ei in terms ofthe pk; to be able to do the inverse, one may rewrite them as

Application to the roots of a polynomialNow view the xi as parameters rather than as variables, and consider the monic polynomial in t with roots x1,…,xn:

where the coefficients are given by the elementary symmetric polynomials in the roots: ak = ek(x1,…,xn). Nowconsider the power sums of the roots

Then according to Newton's identities these can be expressed recursively in terms of the coefficients of thepolynomial using

Application to the characteristic polynomial of a matrixWhen the polynomial above is the characteristic polynomial of a matrix A, the roots are the eigenvalues of the matrix, counted with their algebraic multiplicity. For any positive integer k, the matrix Ak has as eigenvalues the powers xi

k, and each eigenvalue of A contributes its multiplicity to that of the eigenvalue xik of Ak. Then the

coefficients of the characteristic polynomial of Ak are given by the elementary symmetric polynomials in those powers xi

k. In particular, the sum of the xik, which is the k-th power sum ψk of the roots of the characteristic

Newton's identities 62

polynomial of A, is given by its trace:

The Newton identities now relate the traces of the powers Ak to the coefficients of the characteristic polynomial of A.Using them in reverse to express the elementary symmetric polynomials in terms of the power sums, they can beused to find the characteristic polynomial by computing only the powers Ak and their traces.

Relation with Galois theoryFor a given n, the elementary symmetric polynomials ek(x1,…,xn) for k = 1,…, n form an algebraic basis for thespace of symmetric polynomials in x1,…. xn: every polynomial expression in the xi that is invariant under allpermutations of those variables is given by a polynomial expression in those elementary symmetric polynomials, andthis expression is unique up to equivalence of polynomial expressions. This is a general fact known as thefundamental theorem of symmetric polynomials, and Newton's identities provide explicit formulae in the case ofpower sum symmetric polynomials. Applied to the monic polynomial with allcoefficients ak considered as free parameters, this means that every symmetric polynomial expression S(x1,…,xn) inits roots can be expressed instead as a polynomial expression P(a1,…,an) in terms of its coefficients only, in otherwords without requiring knowledge of the roots. This fact also follows from general considerations in Galois theory(one views the ak as elements of a base field, the roots live in an extension field whose Galois group permutes themaccording to the full symmetric group, and the field fixed under all elements of the Galois group is the base field).The Newton identities also permit expressing the elementary symmetric polynomials in terms of the power sumsymmetric polynomials, showing that any symmetric polynomial can also be expressed in the power sums. In factthe first n power sums also form an algebraic basis for the space of symmetric polynomials.

Related identitiesThere is a number of (families of) identities that, while they should be distinguished from Newton's identities, arevery closely related to them.

A variant using complete homogeneous symmetric polynomialsDenoting by hk the complete homogeneous symmetric polynomial that is the sum of all monomials of degree k, thepower sum polynomials also satisfy identities similar to Newton's identities, but not involving any minus signs.Expressed as identities of in the ring of symmetric functions, they read

valid for all k ≥ 1. Contrary to Newton's identities,the left-hand sides do not become zero for large k, and the righthand sides contain ever more nonzero terms. For the first few values of k one has

These relations can be justified by an argument analoguous to the one by comparing coefficients in power seriesgiven above, based in this case on the generating function identity

The other proofs given above of Newton's identities cannot be easily adapted to prove these variants of thoseidentities.

Newton's identities 63

Expressing elementary symmetric polynomials in terms of power sumsA mentioned, Newton's identities can be used to recursively express elementary symmetric polynomials in terms ofpower sums. Doing so requires the introduction of integer denominators, so it can be done in the ring Λ

Q of

symmetric functions with rational coefficients:

and so forth. Applied to a monic polynomial these formulae express the coefficients in terms of the power sums ofthe roots: replace each ei by ai and each pk by ψk.

Expressing complete homogeneous symmetric polynomials in terms of power sumsThe analogous relations involving complete homogeneous symmetric polynomials can be similarly developed,giving equations

and so forth, in which there are only plus signs. These expressions correspond exactly to the cycle index polynomialsof the symmetric groups, if one interprets the power sums pi as indeterminates: the coefficient in the expression forhk of any monomial p1

m1p2m2…pl

ml is equal to the fraction of all permutations of k that have m1 fixed points, m2cycles of length 2, …, and ml cycles of length l. Explicitly, this coefficient can be written as where

; this N is the number permutations commuting with any given permutation π of the givencycle type. The expressions for the elementary symmetric functions have coefficients with the same absolute value,but a sign equal to the sign of π, namely (−1)m2+m4+….

Expressing power sums in terms of elementary symmetric polynomialsOne may also use Newton's identities to express power sums in terms of symmetric polynomials, which does notintroduce denominators:

giving ever longer expressions that do not seem to follow any simple pattern. By consideration of the relations usedto obtain these expressions, it can however be seen that the coefficient of some monomial in theexpression for has the same sign as the coefficient of the corresponding product in the expression for

described above, namely the sign (−1)m2+m4+…. Furthermore the absolute value of the coefficient of M is thesum, over all distinct sequences of elementary symmetric functions whose product is M, of the index of the last onein the sequence: for instance the coefficient of in the expression for p20 will be

, since of all distinct orderings of the five factors e1, onefactor e3 and three factors e4, there are 280 that end with e1, 56 that end with e3, and 168 that end with e4.

Newton's identities 64

Expressing power sums in terms of complete homogeneous symmetric polynomialsFinally one may use the variant identities involving complete homogeneous symmetric polynomials similarly toexpress power sums in term of them:

and so on. Apart from the replacement of each ei by the corresponding hi, the only change with respect to theprevious family of identities is in the signs of the terms, which in this case depend just on the number of factorspresent: the sign of the monomial is −(−1)m1+m2+m3+…. In particular the above description of theabsolute value of the coefficients applies here as well.

Expressions as determinantsOne can obtain explicit formulas for the above expressions in the form of determinants, by considering the first n ofNewton's identities (or it counterparts for the complete homogeneous polynomials) as linear equations in which theelementary symmetric functions are known and the power sums are unknowns (or vice versa), and apply Cramer'srule to find the solution for the final unknown. For instance taking Newton's identities in the form

we consider , , , ..., and as unknowns, and solve for the final one, giving

Solving for instead of for is similar, as the analogous computations for the complete homogeneoussymmetric polynomials; in each case the details are slightly messier than the final results, which are (Macdonald1979, p. 20):

Note that the use of determinants makes that formula for has additional minus signs with respect to the one for , while the situation for the expanded form given earlier is opposite. As remarked in (Littelwood 1950, p. 84) one

Newton's identities 65

can alternatively obtain the formula for by taking the permanent of the matrix for instead of the determinant, andmore generally an expression for any Schur polynomial can be obtained by taking the corresponding immanant ofthis matrix.

Derivation of the identitiesEach of Newton's identities can easily be checked by elementary algebra; however, their validity in general needs aproof. Here are some possible derivations

From the special case n = kOne can obtain the k-th Newton identity in k variables by substitution into

as follows. Substituting xj for t gives

Summing over all j gives

where the terms for i = 0 were taken out of the sum because p0 is (usually) not defined. This equation immediatelygives the k-th Newton identity in k variables. Since this is an identity of symmetric polynomials (homogeneous) ofdegree k, its validity for any number of variables follows from its validity for k variables. Concretely, the identities inn < k variables can be deduced by setting k − n variables to zero. The k-th Newton identity in n > k variables containsmore terms on both sides of the equation than the one in k variables, but its validity will be assured if the coefficientsof any monomial match. Because no individual monomial involves more than k of the variables, the monomial willsurvive the substitution of zero for some set of n − k (other) variables, after which the equality of coefficients is onethat arises in the k-th Newton identity in k (suitably chosen) variables.

Comparing coefficients in seriesA derivation can be given by formal manipulations based on the basic relation

linking roots and coefficients of a monic polynomial. However, to facilitate the manipulations one first "reverses thepolynomials" by substituting 1/t for t and then multiplying both sides by tn to remove negative powers of t, giving

Swapping sides and expressing the ai as the elementary symmetric polynomials they stand for gives the identity

One differentiates both sides with respect to t, and then (for convenience) multiplies by t, to obtain

Newton's identities 66

where the polynomial on the right hand side was first rewritten as a rational function in order to be able to factor outa product from of the summation, then the fraction in the summand was developed as a series in t, and finally thecoefficient of each t j was collected, giving a power sum. (The series in t is a formal power series, but mayalternatively be thought of as a series expansion for t sufficiently close to 0, for those more comfortable with that; infact one is not interested in the function here, but only in the coefficients of the series.) Comparing coefficients of tk

on both sides one obtains

which gives the k-th Newton identity.

As a telescopic sum of symmetric function identitiesThe following derivation, given essentially in (Mead, 1992), is formulated in the ring of symmetric functions forclarity (all identities are independent of the number of variables). Fix some k > 0, and define the symmetric functionr(i) for 2 ≤ i ≤ k as the sum of all distinct monomials of degree k obtained by multiplying one variable raised to thepower i with k − i distinct other variables (this is the monomial symmetric function mγ where γ is a hook shape(i,1,1,…1)). In particular r(k) = pk; for r(1) the description would amount to that of ek, but this case was excludedsince here monomials no longer have any distinguished variable. All products piek−i can be expressed in terms of ther(j) with the first and last case being somewhat special. One has

since each product of terms on the left involving distinct variables contributes to r(i), while those where the variablefrom pi already occurs among the variables of the term from ek−i contributes to r(i + 1), and all terms on the right areso obtained exactly once. For i = k one multiplies by e0 = 1, giving trivially

.Finally the product p1ek−1 for i = 1 gives contributions to r(i + 1) = r(2) like for other values i < k, but the remainingcontributions produce k times each monomial of ek, since any on of the variables may come from the factor p1; thus

.The k-th Newton identity is now obtained by taking the alternating sum of these equations, in which all terms of theform r(i) cancel out.

Newton's identities 67

References• Tignol, Jean-Pierre (2001). Galois' theory of algebraic equations. Singapore: World Scientific.

ISBN 978-981-02-4541-2.• Bergeron, F.; Labelle, G.; and Leroux, P. (1998). Combinatorial species and tree-like structures. Cambridge:

Cambridge University Press. ISBN 978-0-521-57323-8.• Cameron, Peter J. (1999). Permutation Groups. Cambridge: Cambridge University Press.

ISBN 978-0-521-65378-7.• Cox, David; Little, John, and O'Shea, Donal (1992). Ideals, Varieties, and Algorithms. New York:

Springer-Verlag. ISBN 978-0-387-97847-5.• Eppstein, D.; Goodrich, M. T. (2007). "Space-efficient straggler identification in round-trip data streams via

Newton's identities and invertible Bloom filters". Algorithms and Data Structures, 10th International Workshop,WADS 2007. Springer-Verlag, Lecture Notes in Computer Science 4619. pp. 637–648. arXiv:0704.3313

• Littlewood, D. E. (1950). The theory of group characters and matrix representations of groups. Oxford: OxfordUniversity Press. viii+310. ISBN 0-8218-4067-3.

• Macdonald, I. G. (1979). Symmetric functions and Hall polynomials. Oxford Mathematical Monographs. Oxford:The Clarendon Press, Oxford University Press. viii+180. ISBN 0-19-853530-9. MR84g:05003.

• Macdonald, I. G. (1995). Symmetric functions and Hall polynomials. Oxford Mathematical Monographs (Seconded.). New York: Oxford Science Publications. The Clarendon Press, Oxford University Press. p. x+475.ISBN 0-19-853489-2. MR96h:05207.

• Mead, D.G. (1992-10). "Newton's Identities". The American Mathematical Monthly (Mathematical Association ofAmerica) 99 (8): 749–751. doi:10.2307/2324242. JSTOR 2324242.

• Stanley, Richard P. (1999). Enumerative Combinatorics, Vol. 2. Cambridge University Press.ISBN 0-521-56069-1 (hardback), ISBN 0-521-78987-7 (paperback).

• Sturmfels, Bernd (1992). Algorithms in Invariant Theory. New York: Springer-Verlag. ISBN 978-0-387-82445-1.• Tucker, Alan (1980). Applied Combinatorics (5/e ed.). New York: Wiley. ISBN 978-0-471-73507-6.

External links• Newton–Girard formulas [1] on MathWorld• A Matrix Proof of Newton's Identities [2] in Mathematics Magazine

References[1] http:/ / mathworld. wolfram. com/ Newton-GirardFormulas. html[2] http:/ / links. jstor. org/ sici?sici=0025-570X(200010)73%3A4%3C313%3AAMPONI%3E2. 0. CO%3B2-0

Polynomial ring 68

Polynomial ringIn mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set ofpolynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much ofmathematics, from the Hilbert basis theorem, to the construction of splitting fields, and to the understanding of alinear operator. Many important conjectures involving polynomial rings, such as Serre's problem, have influencedthe study of other rings, and have influenced even the definition of other rings, such as group rings and rings offormal power series.

Polynomials in one variable over a field

PolynomialsA polynomial in X with coefficients in a field K is an expression of the form

where p0, …, pm are elements of K, the coefficients of p, and X, X 2, … are formal symbols ("the powers of X").Such expressions can be added and multiplied, and then brought into the same form using the ordinary rules formanipulating algebraic expressions, such as associativity, commutativity, distributivity, and collecting the similarterms. Any term pkX k with zero coefficient, pk = 0, may be omitted. The product of the powers of X is defined by thefamiliar formula

where k and l are any natural numbers. Two polynomials are considered to be equal if and only if the correspondingcoefficients for each power of X are equal. By convention, X 1 = X, X 0 = 1, and the sum defining the polynomial pmay be viewed as the linear combination of the symbols X m, …, X 1, X 0 with coefficients pm, …, p1, p0. Using thesummation symbol, the same polynomial is expressed more compactly as follows:

The summation limits are frequently omitted, so that the same polynomial is written as

and it is understood that only finitely many terms are present, i.e. pk is zero for all large enough values of k, in ourcase, for k > m. The degree of a polynomial is the largest k such that the coefficient of X k is not zero. In the specialcase of zero polynomial, all of whose coefficients are zero, the degree is undefined, or sometimes defined to be thesymbol −∞.[1]

The polynomial ring K[X]The set of all polynomials with coefficients in the field K forms a commutative ring denoted K[X] and is called thering of polynomials over K. The symbol X is commonly called the "variable", and this ring is also called the ring ofpolynomials in one variable over K, to distinguish it from more general rings of polynomials in several variables.This terminology is suggested by the important cases of polynomials with real or complex coefficients, which maybe alternatively viewed as real or complex polynomial functions. However, in general, X and its powers, X k, aretreated as formal symbols, not as elements of the field K. One can think of the ring K[X] as arising from K by addingone new element X that is external to K and requiring that X commute with all elements of K. In order for K[X] toform a ring, all powers of X have to be included as well, and this leads to the definition of polynomials as linearcombinations of the powers of X with coefficients in K.

Polynomial ring 69

A ring has two binary operations, addition and multiplication. In the case of the polynomial ring K[X], theseoperations are explicitly given by the following formulas:

and

In the first formula, one of the polynomials may be extended by adding "dummy terms" with zero coefficients, sothat the same set of powers formally appears in both summands. In the second formula, the inner summation in theright hand side is only extended over indices within bounds, 0 ≤ i ≤ m and 0 ≤ j ≤ n. Alternative forms of expressingaddition and multiplication, without using explicit bounds in the sums, are as follows:

and

Since only finitely many coefficients ai and bj are non-zero, all sums in effect have only finitely many terms, andhence represent polynomials from K[X].Since a polynomial from K[X] can be multiplied by a "scalar" k from K to yield a new polynomial, K[X] actuallyconstitute an associative algebra over K. Viewed as a vector space, K[X] has a basis consisting of the countablyinfinite set {1, X, X 2, X 3, ...}.More generally, the field K can be replaced by any commutative ring R, giving rise to the polynomial ring over R ,which is denoted R[X].

Properties of K[X]The polynomial ring K[X] is remarkably similar to the ring Z of integers in many respects. This analogy and thearithmetic of the ring of polynomials were thoroughly investigated by Gauss and his theory served as a model fordevelopment of abstract algebra in the second half of the nineteenth century in the works of Kummer, Kronecker,and Dedekind.

K[X] is an integral domain

The first property of the polynomial ring is elementary and says that a product of two non-zero polynomials is also anon-zero polynomial. Indeed, the product of a polynomial p of degree m starting with pmX m, pm ≠ 0, and apolynomial q of degree n starting with qnX n, qn ≠ 0, is the polynomial pq starting with the term rX m+n, where thecoefficient r = pmqn ≠ 0. Hence pq is a non-zero polynomial of degree m + n. Commutative rings with unity e=x0 inwhich the product of any two non-zero elements is non-zero are called integral domains, and thus the polynomialring K[X] is an integral domain.

Polynomial ring 70

Factorization in K[X]

The next property of the polynomial ring is much deeper. Already Euclid noted that every positive integer can beuniquely factored into a product of primes — this statement is now called the fundamental theorem of arithmetic.The proof is based on Euclid's algorithm for finding the greatest common divisor of natural numbers. At each step ofthis algorithm, a pair (a, b), a > b, of natural numbers is replaced by a new pair (b, r), where r is the remainder fromthe division of a by b, and the new numbers are smaller. Gauss remarked that the procedure of division with theremainder can also be defined for polynomials: given two polynomials p and q, where q ≠ 0, one can write

where the quotient u and the remainder r are polynomials, the degree of r is less than the degree of q, and adecomposition with these properties is unique. The quotient and the remainder are found using the polynomial longdivision. The degree of the polynomial now plays a role similar to the absolute value of an integer: it is strictly lessin the remainder r than it is in q, and when repeating this step such decrease cannot go on indefinitely. Thereforeeventually some division will be exact, at which point the last non-zero remainder is the greatest common divisor ofthe initial two polynomials. Using the existence of greatest common divisors, Gauss was able to simultaneouslyrigorously prove the fundamental theorem of arithmetic for integers and its generalization to polynomials. In factthere exist other commutative rings than Z and K[X] that similarly admit an analogue of the Euclidean algorithm; allsuch rings are called Euclidean rings. Rings for which there exists unique (in an appropriate sense) factorization ofnonzero elements into irreducible factors are called unique factorization domains or factorial rings; the givenconstruction shows that all Euclidean rings, and in particular Z and K[X], are unique factorization domains.Another corollary of the polynomial division with the remainder is the fact that every proper ideal I of K[X] isprincipal, i.e. I consists of the multiples of a single polynomial f. Thus the polynomial ring K[X] is a principal idealdomain, and for the same reason every Euclidean domain is a principal ideal domain. Also every principal idealdomain is a unique-factorization domain. These deductions make essential use of the fact that the polynomialcoefficients lie in a field, namely in the polynomial division step, which requires the leading coefficient of q, whichis only known to be non-zero, to have an inverse. If R is an integral domain that is not a field then R[X] is neither aEuclidean domain nor a principal ideal domain; however it could still be a unique factorization domain (and will beso if and only it R itself is a unique factorization domain, for instance if it is Z or another polynomial ring).

Quotient ring of K[X]

The ring K[X] of polynomials over K is obtained from K by adjoining one element, X. It turns out that anycommutative ring L containing K and generated as a ring by a single element in addition to K can be described usingK[X]. In particular, this applies to finite field extensions of K.Suppose that a commutative ring L contains K and there exists an element θ of L such that the ring L is generated byθ over K. Thus any element of L is a linear combination of powers of θ with coefficients in K. Then there is a uniquering homomorphism φ from K[X] into L which does not affect the elements of K itself (it is the identity map on K)and maps each power of X to the same power of θ. Its effect on the general polynomial amounts to "replacing X withθ":

By the assumption, any element of L appears as the right hand side of the last expression for suitable m and elementsa0, …, am of K. Therefore, φ is surjective and L is a homomorphic image of K[X]. More formally, let Ker φ be thekernel of φ. It is an ideal of K[X] and by the first isomorphism theorem for rings, L is isomorphic to the quotient ofthe polynomial ring K[X] by the ideal Ker φ. Since the polynomial ring is a principal ideal domain, this ideal isprincipal: there exists a polynomial p∈K[X] such that

Polynomial ring 71

A particularly important application is to the case when the larger ring L is a field. Then the polynomial p must beirreducible. Conversely, the primitive element theorem states that any finite separable field extension L/K can begenerated by a single element θ∈L and the preceding theory then gives a concrete description of the field L as thequotient of the polynomial ring K[X] by a principal ideal generated by an irreducible polynomial p. As an illustration,the field C of complex numbers is an extension of the field R of real numbers generated by a single element i suchthat i2 + 1 = 0. Accordingly, the polynomial X2 + 1 is irreducible over R and

More generally, given a (not necessarily commutative) ring A containing K and an element a of A that commuteswith all elements of K, there is a unique ring homomorphism from the polynomial ring K[X] to A that maps X to a:

This homomorphism is given by the same formula as before, but it is not surjective in general. The existence anduniqueness of such a homomorphism φ expresses a certain universal property of the ring of polynomials in onevariable and explains ubiquity of polynomial rings in various questions and constructions of ring theory andcommutative algebra.

The polynomial ring in several variables

PolynomialsA polynomial in n variables X1,…, Xn with coefficients in a field K is defined analogously to a polynomial in onevariable, but the notation is more cumbersome. For any multi-index α = (α1,…, αn), where each αi is a non-negativeinteger, let

The product Xα is called the monomial of multidegree α. A polynomial is a finite linear combination of monomialswith coefficients in K

and only finitely many coefficients pα are different from 0. The degree of a monomial Xα, frequently denoted |α|, isdefined as

and the degree of a polynomial p is the largest degree of a monomial occurring with non-zero coefficient in theexpansion of p.

The polynomial ringPolynomials in n variables with coefficients in K form a commutative ring denoted K[X1,…, Xn], or sometimes K[X],where X is a symbol representing the full set of variables, X = (X1,…, Xn), and called the polynomial ring in nvariables. The polynomial ring in n variables can be obtained by repeated application of K[X] (the order by which isirrelevant). For example, K[X1, X2] is isomorphic to K[X1][X2]. This ring plays fundamental role in algebraicgeometry. Many results in commutative and homological algebra originated in the study of its ideals and modulesover this ring.A polynomial ring with coefficients in is the free commutative ring over its set of variables.

Polynomial ring 72

Hilbert's NullstellensatzA group of fundamental results concerning the relation between ideals of the polynomial ring K[X1,…, Xn] andalgebraic subsets of Kn originating with David Hilbert is known under the name Nullstellensatz (literally:"zero-locus theorem").• (Weak form, algebraically closed field of coefficients). Let K be an algebraically closed field. Then every maximal

ideal m of K[X1,…, Xn] has the form

• (Weak form, any field of coefficients). Let k be a field, K be an algebraically closed field extension of k, and I bean ideal in the polynomial ring k[X1,…, Xn]. Then I contains 1 if and only if the polynomials in I do not have anycommon zero in Kn.

• (Strong form). Let k be a field, K be an algebraically closed field extension of k, I be an ideal in the polynomialring k[X1,…, Xn],and V(I) be the algebraic subset of Kn defined by I. Suppose that f is a polynomial whichvanishes at all points of V(I). Then some power of f belongs to the ideal I:

Using the notion of the radical of an ideal, the conclusion says that f belongs to the radical of I. As a corollaryof this form of Nullstellensatz, there is a bijective correspondence between the radical ideals of K[X1,…, Xn]for an algebraically closed field K and the algebraic subsets of the n-dimensional affine space Kn. It arisesfrom the map

The prime ideals of the polynomial ring correspond to irreducible subvarieties of Kn.

Properties of the ring extension R ⊂ R[X]One of the basic techniques in commutative algebra is to relate properties of a ring with properties of its subrings.The notation R ⊂ S indicates that a ring R is a subring of a ring S. In this case S is called an overring of R and onespeaks of a ring extension. This works particularly well for polynomial rings and allows one to establish manyimportant properties of the ring of polynomials in several variables over a field, K[X1,…, Xn], by induction in n.

Summary of the resultsIn the following properties, R is a commutative ring and S = R[X1,…, Xn] is the ring of polynomials in n variablesover R. The ring extension R ⊂ S can be built from R in n steps, by successively adjoining X1,…, Xn. Thus toestablish each of the properties below, it is sufficient to consider the case n = 1.• If R is an integral domain then the same holds for S.• If R is a unique factorization domain then the same holds for S. The proof is based on the Gauss lemma.• Hilbert's basis theorem: If R is a Noetherian ring, then the same holds for S.• Suppose that R is a Noetherian ring of finite global dimension. Then

An analogous result holds for Krull dimension.

Polynomial ring 73

GeneralizationsPolynomial rings have been generalized in a great many ways, including polynomial rings with generalizedexponents, power series rings, noncommutative polynomial rings, and skew-polynomial rings.

Infinitely many variablesThe possibility to allow an infinite set of indeterminates is not really a generalization, as the ordinary notion ofpolynomial ring allows for it. It is then still true that each monomial involves only a finite number of indeterminates(so that its degree remains finite), and that each polynomial is a linear combination of monomials, which bydefinition involves only finitely many of them. This explains why such polynomial rings are relatively seldomconsidered: each individual polynomial involves only finitely many indeterminates, and even any finite computationinvolving polynomials remains inside some subring of polynomials in finitely many indeterminates.In the case of infinitely many indeterminates, one can consider a ring strictly larger than the polynomial ring butsmaller than the power series ring, by taking the subring of the latter formed by power series whose monomials havea bounded degree. Its elements still have a finite degree and are therefore are somewhat like polynomials, but it ispossible for instance to take the sum of all indeterminates, which is not a polynomial. A ring of this kind plays a rolein constructing the ring of symmetric functions.

Generalized exponentsA simple generalization only changes the set from which the exponents on the variable are drawn. The formulas foraddition and multiplication make sense as long as one can add exponents: Xi·Xj = Xi+j. A set for which additionmakes sense (is closed and associative) is called a monoid. The set of functions from a monoid N to a ring R whichare nonzero at only finitely many places can be given the structure of a ring known as R[N], the monoid ring of Nwith coefficients in R. The addition is defined component-wise, so that if c = a+b, then cn = an + bn for every n in N.The multiplication is defined as the Cauchy product, so that if c = a·b, then for each n in N, cn is the sum of all aibjwhere i, j range over all pairs of elements of N which sum to n.When N is commutative, it is convenient to denote the function a in R[N] as the formal sum:

and then the formulas for addition and multiplication are the familiar:

and

where the latter sum is taken over all i, j in N that sum to n.Some authors such as (Lang 2002, II,§3) go so far as to take this monoid definition as the starting point, and regularsingle variable polynomials are the special case where N is the monoid of non-negative integers. Polynomials inseveral variables simply take N to be the direct product of several copies of the monoid of non-negative integers.Several interesting examples of rings and groups are formed by taking N to be the additive monoid of non-negativerational numbers, (Osbourne 2000, §4.4).

Polynomial ring 74

Power seriesPower series generalize the choice of exponent in a different direction by allowing infinitely many nonzero terms.This requires various hypotheses on the monoid N used for the exponents, to ensure that the sums in the Cauchyproduct are finite sums. Alternatively, a topology can be placed on the ring, and then one restricts to convergentinfinite sums. For the standard choice of N, the non-negative integers, there is no trouble, and the ring of formalpower series is defined as the set of functions from N to a ring R with addition component-wise, and multiplicationgiven by the Cauchy product. The ring of power series can be seen as the completion of the polynomial ring.

Noncommutative polynomial ringsFor polynomial rings of more than one variable, the products X·Y and Y·X are simply defined to be equal. A moregeneral notion of polynomial ring is obtained when the distinction between these two formal products is maintained.Formally, the polynomial ring in n noncommuting variables with coefficients in the ring R is the monoid ring R[N],where the monoid N is the free monoid on n letters, also known as the set of all strings over an alphabet of nsymbols, with multiplication given by concatenation. Neither the coefficients nor the variables need commuteamongst themselves, but the coefficients and variables commute with each other.Just as the polynomial ring in n variables with coefficients in the commutative ring R is the free commutativeR-algebra of rank n, the noncommutative polynomial ring in n variables with coefficients in the commutative ring Ris the free associative, unital R-algebra on n generators, which is noncommutative when n > 1.

Differential and skew-polynomial ringsOther generalizations of polynomials are differential and skew-polynomial rings.A differential polynomial ring is formed from a ring R and a derivation δ of R into R. Then the multiplication isextended from the relation X·a = a·X + δ(a). The standard example, called a Weyl algebra, takes R to be apolynomial ring k[t], and X to be the standard polynomial derivative . One views the elements of R[X] asdifferential operators on the polynomial ring k[t], with elements f(t) of R=k[t] acting as multiplication, and X actingas the derivative in t. Labelling t = Y, one gets the canonical commutation relation, X·Y − Y·X = 1, making the ringexplicitly a Weyl algebra. This is a fundamentally important ring, (Lam 2001, §1,ex1.9).The skew-polynomial ring is defined for a ring R and a ring endomorphism f of R, multiplication is extended fromthe relation X·r = f(r)·X to give an associative multiplication that distributes over the standard addition. Moregenerally, one has a homomorphism F from the monoid N into the endomorphism ring of R, and Xn·r = F(n)(r)·Xn, asin (Lam 2001, §1,ex 1.11). Skew polynomial rings are closely related to crossed product algebras.

References[1] P.M.Cohn Algebra Vol 1, Wiley 1974, p.127 ISBN 0-471-16430-5

• Lam, Tsit-Yuen (2001), A First Course in Noncommutative Rings, Berlin, New York: Springer-Verlag,ISBN 978-0-387-95325-0

• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York:Springer-Verlag, ISBN 978-0-387-95385-4, MR1878556

• Osborne, M. Scott (2000), Basic homological algebra, Graduate Texts in Mathematics, 196, Berlin, New York:Springer-Verlag, ISBN 978-0-387-98934-1, MR1757274

Quantum invariant 75

Quantum invariantIn the mathematical field of knot theory, a quantum invariant of a knot or link is a linear sum of colored Jonespolynomial of surgery presentations of the knot complement.[1]

List of invariants• Finite type invariant• Kontsevich invariant• Kashaev's invariant• Witten–Reshetikhin–Turaev invariant (Chern–Simons)• Invariant differential operator[2]

• Rozansky–Witten invariant• Vassiliev knot invariant• Dehn invariant• LMO invariant [3]

• Turaev–Viro invariant• Dijkgraaf–Witten invariant [4]

• Reshetikhin–Turaev invariant• Tau-invariant• I-Invariant• Klein J-invariant• Quantum isotopy invariant [5]

• Ermakov–Lewis invariant• Hermitian invariant• Goussarov–Habiro theory of finite-type invariant• Linear quantum invariant (orthogonal function invariant)• Murakami–Ohtsuki TQFT• Generalized Casson invariant• Casson-Walker invariant• Khovanov–Rozansky invariant• HOMFLY polynomial quantum invariant[6]

• K-theory invariants• Atiyah–Patodi–Singer eta invariant• Link invariant [7]

• Casson invariant• Seiberg–Witten invariant• Gromov–Witten invariant• Arf invariant• Hopf invariant

Quantum invariant 76

References[1] http:/ / projecteuclid. org/ DPubS/ Repository/ 1. 0/ Disseminate?view=body& id=pdf_1& handle=euclid. ojm/ 1183667985[2] http:/ / arxiv. org/ abs/ math. QA/ 0406194[3] http:/ / arxiv. org/ abs/ math/ 0009222v1[4] http:/ / hal. archives-ouvertes. fr/ docs/ 00/ 09/ 02/ 99/ PDF/ equality_arxiv_1. pdf[5] http:/ / knot. kaist. ac. kr/ 7thkgtf/ Lawton1. pdf[6] http:/ / www. hausdorff-research-institute. uni-bonn. de/ geometry-and-physics-seminars[7] http:/ / www. springerlink. com/ content/ u416971m947560r7/ fulltext. pdf

Further reading• Reference to Frank Quinn: Freedman, Michael H.; Quinn, Frank (1990). Topology of 4-Manifolds. Princeton:

Princeton University Press. ISBN 0691085773.• Reshetikhin, N. & Turaev, V. (1991). "Invariants of 3-manifolds via link polynomials and quantum groups".

Invent. Math. 103 (1): 547–597. doi:10.1007/BF01239527.• Kontsevich, Maxim (1993). "Vassiliev's knot invariants". Adv. Soviet Math. 16: 137–150.• Ohtsuki, Tomotada (2002). Quantum Invariants: A Study of Knots, 3-Manifolds, and their Sets. Series on Knots

and Everything. 29. Singapore: World Scientific Publishing. ISBN 9810246757.

External links• Quantum invariants of knots and 3-manifolds By Vladimir G. Turaev (http:/ / books. google. com/

books?id=yQRDNCJ0iOUC& pg=PA137& lpg=PA137& dq="quantum+ invariant"& source=bl&ots=CeAUzYzr2D& sig=GIThkNK4Rke2dqo74VN4poLScR8& hl=en& ei=MnX2SdzIB6KUMumIwcMP&sa=X& oi=book_result& ct=result& resnum=9#PPP1,M1)

Radical polynomialIn mathematics, in the realm of abstract algebra, a radical polynomial is a multivariate polynomial over a field thatcan be expressed as a polynomial in the sum of squares of the variables. That is, if

is a polynomial ring, the ring of radical polynomials is the subring generated by the polynomial

Radical polynomials are characterized as precisely those polynomials that are invariant under the action of theorthogonal group.The ring of radical polynomials is a graded subalgebra of the ring of all polynomials.The standard separation of variables theorem asserts that every polynomial can be expressed as a finite sum of terms,each term being a product of a radical polynomial and a harmonic polynomial. This is equivalent to the statementthat the ring of all polynomials is a free module over the ring of radical polynomials.

Reynolds operator 77

Reynolds operatorIn fluid dynamics and invariant theory, a Reynolds operator is a mathematical operator given by averagingsomething over a group action, that satisfies a set of properties called Reynolds rules. In fluid dynamics Reynoldsoperators are often encountered in models of turbulent flows, particularly the Reynolds-averaged Navier-Stokesequations, where the average is typically taken over the fluid flow under the group of time translations. In invarianttheory the average is often taken over a compact group or reductive algebraic group acting on a commutativealgebra, such as a ring of polynomials. Reynolds operators were introduced into fluid dynamics by OsbourneReynolds (1895) and named by J. Kampé de Fériet (1934, 1935, 1949).

DefinitionReynolds operators are used in fluid dynamics, functional analysis, and invariant theory, and the notation anddefinitions in these areas differ slightly. A Reynolds operator acting on φ is sometimes denoted by R(φ), P(φ), ρ(φ),〈φ〉, or φ. Reynolds operators are usually linear operators acting on some algebra of functions, satisfying theidentity

R(R(φ)ψ) = R(φ)R(ψ) for all φ, ψand possibly some other conditions, such as commuting with various group actions.

Invariant theoryIn invariant theory a Reynolds operator R is usually a linear operator satisfying

R(R(φ)ψ) = R(φ)R(ψ) for all φ, ψand

R(1) =1.Together these conditions imply that R is idempotent: R2=R. The Reynolds operator will also usually commute withsome group action, and project onto the invariant elements of this group action.

Functional analysisIn functional analysis a Reynolds operator is a linear operator R acting on some algebra of functions φ, satisfying theReynolds identity

R(φψ) = R(φ)R(ψ) +R((φ−R(φ))(ψ−R(ψ))) for all φ, ψThe operator R is called an averaging operator if it is linear and satisfies

R(R(φ)ψ) = R(φ)R(ψ) for all φ, ψ.If R(R(φ)) = R(φ) for all φ then R is an averaging operator if and only if it is a Reynolds operator. Sometimes theR(R(φ)) = R(φ) condition is added to the definition of Reynolds operators.

Reynolds operator 78

Fluid dynamics

Let and be two random variables, and be an arbitrary constant. Then the properties satisfied by Reynoldsoperators, for an operator include linearity and the averaging property:

which implies In addition the Reynolds operator is often assumed to commute with space and time translations:

Any operator satisfying these properties is a Reynolds operator.[1]

ExamplesReynolds operators are often given by projecting onto an invariant subspace of a group action.• The "Reynolds operator" considered by Reynolds (1895) was essentially the projection of a fluid flow to the

"average" fluid flow, which can be thought of as projection to time-invariant flows. Here the group action is givenby the action of the group of time-translations.

• Suppose that G is a reductive algebraic group or a compact group, and V is a finite-dimensional representation ofG. Then G also acts on the symmetric algebra SV of polynomials. The Reynolds operator R is the G-invariantprojection from SV to the subring SVG of elements fixed by G.

References[1] Sagaut, Pierre (2006). Large Eddy Simulation for Incompressible Flows (Third Edition ed.). Springer. ISBN 3540263446.

• Kampé de Fériet, J. (1934), La Science Aérienne 3: 9–34• Kampé de Fériet, J. (1935), La Science Aérienne 4: 12–52• Kampé de Fériet, J. (1949), "Sur un problème d'algèbre abstraite posé par la définition de la moyenne dans la

théorie de la turbulence", Annales de la Societé Scientifique de Bruxelles. Série I. Sciences Mathématiques,Astronomiques et Physiques 63: 165–180, ISSN 0037-959X, MR0032718

• Reynolds, O. (1895), "On the dynamical theory of incompressible viscous fluids and the determination of thecriterion", Philos. Trans. Roy. Soc. Ser. A 186: 123–164, JSTOR 90643

• Rota, Gian-Carlo (2003), Gian-Carlo Rota on analysis and probability, Contemporary Mathematicians, Boston,MA: Birkhäuser Boston, ISBN 978-0-8176-4275-4, MR1944526 Reprints several of Rota's papers on Reynoldsoperators, with commentary.

• Rota, Gian-Carlo (1964), "Reynolds operators", Proc. Sympos. Appl. Math., Vol. XVI, Providence, R.I.: Amer.Math. Soc., pp. 70–83, MR0161140

• Sturmfels, Bernd (1993), Algorithms in invariant theory, Texts and Monographs in Symbolic Computation,Berlin, New York: Springer-Verlag, ISBN 978-3-211-82445-0, MR1255980

Riemann invariant 79

Riemann invariantRiemann invariants are mathematical transformations made on a a system of quasi-linear first order partialdifferential equations to make them more easily solvable. Riemann invariants are constant along the characteristiccurves of the partial differential equations where they obtain the name invariant. They were first obtained byBernhard Riemann in his work on plane waves in gas dynamics.[1]

Mathematical theoryConsider the set of hyperbolic partial differential equations of the form

where and are the elements of the matrices and where and are elements of vectors. It will beasked if it is possible to rewrite this equation to

To do this curves will be introduced in the plane defined by the vector field . The term in the bracketswill be rewritten in terms of a total derivative where are parametrized as

comparing the last two equations we find

which can be now written in characteristic form

where we must have the conditions

, where can be eliminated to give the necessary condition

so for a nontrival solution is the determinant

For Riemann invariants we are concerned with the case when the matrix is an identity matrix to form

notice this is homogeneous due to the vector being zero. In characteristic form the system is

with

Where is the left eigenvector of the matrix and is the characteristic speeds of the eigenvalues of thematrix which satisfy

To simplify these characteristic equations we can make the transformations such that

which form

Riemann invariant 80

An integrating factor can be multiplied in to help integrate this. So the system now has the characteristic form

on

which is equivalent to the diagonal system[2]

The solution of this system can be given by the generalized hodograph method.[3] [4]

ExampleConsider the shallow water equations

write this system in matrix form

where the matrix from the analysis above the eigenvalues and eigenvectors need to be found.The eigenvalues arefound to satisfy

to give

and the eigenvectors are found to be

where the riemann invariants are

In shallow water equations there is the relation to give the riemann invariants

to give the equations

Which can be solved by the hodograph transformation. If the matrix form of the system of pde's is in the form

Then it may be possible to multiply across by the inverse matrix so long as the matrix determinant of is notzero.

Riemann invariant 81

References[1] B. Riemann (1860) url=http:www.maths.tcd.ie/pub/HistMath/People/Riemann/Welle/Welle.pdf[2] Whitham, G. B. (1974). Linear and Nonlinear Waves. Wiley. ISBN 978-0471940906.[3] Kamchatnov, A. M. (2000). Nonlinear Periodic Waves and their Modulations. World Scientific. ISBN 978-9810244071.[4] Tsarev, S. P. (1985). "On Poisson brackets and one-dimensional hamiltonian systems of hydrodynamic type" (http:/ / ikit. institute. sfu-kras.

ru/ files/ ikit/ Tsarev-DAN-1985-eng. pdf). Soviet Mathematics Doklady 31 (3): 488–491. MR87b:58030. Zbl 0605.35075. .

Ring of symmetric functionsIn algebra and in particular in algebraic combinatorics, the ring of symmetric functions, is a specific limit of therings of symmetric polynomials in n indeterminates, as n goes to infinity. This ring serves as universal structure inwhich relations between symmetric polynomials can be expressed in a way independent of the number n ofindeterminates (but its elements are neither polynomials nor functions). Among other things, this ring plays animportant role in the representation theory of the symmetric groups.

Symmetric polynomialsThe study of symmetric functions is based on that of symmetric polynomials. In a polynomial ring in some finite setof indeterminates, there is an action by ring automorphisms of the symmetric group on (the indices of) theindeterminates (simultaneaously substituting each of them for another according to the permutation used). Theinvariants for this action form the subring of symmetric polynomials. If the indeterminates are X1,…,Xn, thenexamples of such symmetric polynomials are

and

A somewhat more complicated example is X13X2X3 +X1X2

3X3 +X1X2X33 +X1

3X2X4 +X1X23X4 +X1X2X4

3 +… wherethe summation goes on to include all products of the third power of some variable and two other variables. There aremany specific kinds of symmetric polynomials, such as elementary symmetric polynomials, power sum symmetricpolynomials, monomial symmetric polynomials, complete homogeneous symmetric polynomials, and Schurpolynomials.

The ring of symmetric functionsMost relations between symmetric polynomials do not depend on the number n of indeterminates, other than thatsome polynomials in the relation might require n to be large enough in order to be defined. For instance the Newton'sidentity for the third power sum polynomial leads to

where the denote elementary symmetric polynomials; this formula is valid for all natural numbers n, and the onlynotable dependency on it is that ek(X1,…,Xn) = 0 whenever n < k. One would like to write this as an identityp3 = e1

3 − 3e2e1 + 3e3 that does not depend on n at all, and this can be done in the ring of symmetric polynomials. Inthat ring there are elements ek for all integers k ≥ 1, and an arbitrary element can be given by a polynomialexpression in them.

Ring of symmetric functions 82

DefinitionsA ring of symmetric polynomials can be defined over any commutative ring R, and will be denoted ΛR; the basiccase is for R = Z. The ring ΛR is in fact a graded R-algebra. There are two main constructions for it; the first onegiven below can be found in (Stanley, 1999), and the second is essentially the one given in (Macdonald, 1979).

As a ring of formal power series

The easiest (though somewhat heavy) construction starts with the ring of formal power seriesR''X''₁,''X''₂,… over R in infinitely many indeterminates; one defines ΛR as its subringconsisting of power series S that satisfy1. S is invariant under any permutation of the indeterminates, and2. the degrees of the monomials occurring in S are bounded.Note that because of the second condition, power series are used here only to allow infinitely many terms of a fixeddegree, rather than to sum terms of all possible degrees. Allowing this is necessary because an element that containsfor instance a term X1 should also contain a term Xi for every i > 1 in order to be symmetric. Unlike the whole powerseries ring, the subring ΛR is graded by the total degree of monomials: due to condition 2, every element of ΛR is afinite sum of homogeneous elements of ΛR (which are themselves infinite sums of terms of equal degree). For everyk ≥ 0, the element ek ∈ ΛR is defined as the formal sum of all products of k distinct indeterminates, which is clearlyhomogeneous of degree k.

As an algebraic limit

Another construction of ΛR takes somewhat longer to describe, but better indicates the relationship with the ringsR[X1,…,Xn]Sn of symmetric polynomials in n indeterminates. For every n there is a surjective ring homomorphismρn from the analoguous ring R[X1,…,Xn+1]Sn+1 with one more indeterminate onto R[X1,…,Xn]Sn, defined by settingthe last indeterminate Xn+1 to 0. Although ρn has a non-trivial kernel, the nonzero elements of that kernel have degreeat least (they are multiples of X1X2…Xn+1). This means that the restriction of ρn to elements of degree atmost n is a bijective linear map, and ρn(ek(X1,…,Xn+1)) = ek(X1,…,Xn) for all k ≤ n. The inverse of this restrictioncan be extended uniquely to a ring homomorphism φn from R[X1,…,Xn]Sn to R[X1,…,Xn+1]Sn+1, as follows forinstance from the fundamental theorem of symmetric polynomials. Since the imagesφn(ek(X1,…,Xn)) = ek(X1,…,Xn+1) for k = 1,…,n are still algebraically independent over R, the homomorphism φn isinjective and can be viewed as a (somewhat unusual) inclusion of rings. The ring ΛR is then the "union" (direct limit)of all these rings subject to these inclusions. Since all φn are compatible with the grading by total degree of the ringsinvolved, ΛR obtains the structure of a graded ring.This construction differs slightly from the one in (Macdonald, 1979). That construction only uses the surjectivemorphisms ρn without mentioning the injective morphisms φn: it constructs the homogeneous components of ΛRseparately, and equips their direct sum with a ring structure using the ρn. It is also observed that the result can bedescribed as an inverse limit in the category of graded rings. That description however somewhat obscures animportant property typical for a direct limit of injective morphisms, namely that every individual element(symmetric function) is already faithfully represented in some object used in the limit construction, here a ringR[X1,…,Xd]Sd. It suffices to take for d the degree of the symmetric function, since the part in degree d is of that ringis mapped isomorphically to rings with more indeterminates by φn for all n ≥ d. This implies that for studyingrelations between individual elements, there is no fundamental difference between symmetric polynomials andsymmetric functions.

Ring of symmetric functions 83

Defining individual symmetric functionsIt should be noted that the name "symmetric function" for elements of ΛR is a misnomer: in neither construction theelements are functions, and in fact, unlike symmetric polynomials, no function of independent variables can beassociated to such elements (for instance e1 would be the sum of all infinitely many variables, which is not definedunless restrictions are imposed on the variables). However the name is traditional and well established; it can befound both in (Macdonald, 1979), which says (footnote on p. 12)

The elements of Λ (unlike those of Λn) are no longer polynomials: they are formal infinite sums ofmonomials. We have therefore reverted to the older terminology of symmetric functions.

(here Λn denotes the ring of symmetric polynomials in n indeterminates), and also in (Stanley, 1999).To define a symmetric function one must either indicate directly a power series as in the first construction, or give asymmetric polynomial in n indeterminates for every natural number n in a way compatible with the secondconstruction. An expression in an unspecified number of indeterminates may do both, for instance

can be taken as the definition of an elementary symmetric function if the number of indeterminates is infinite, or asthe definition of an elementary symmetric polynomial in any finite number of indeterminates. Symmetricpolynomials for the same symmetric function should be compatible with the morphisms ρn (decreasing the numberof indeterminates is obtained by setting some of them to zero, so that the coefficients of any monomial in theremaining indeterminates is unchanged), and their degree should remain bounded. (An example of a family ofsymmetric polynomials that fails both conditions is ; the family fails only the secondcondition.) Any symmetric polynomial in n indeterminates can be used to construct a compatible family ofsymmetric polynomials, using the morphisms ρi for i < n to decrease the number of indeterminates, and φi for i ≥ nto increase the number of indeterminates (which amounts to adding all monomials in new indeterminates obtained bysymmetry from monomials already present).The following are fundamental examples of symmetric functions.• The monomial symmetric functions mα, determined by monomial Xα (where α = (α1,α2,…) is a sequence of

natural numbers); mα is the sum of all monomials obtained by symmetry from Xα. For a formal definition,consider such sequences to be infinite by appending zeroes (which does not alter the monomial), and define therelation "~" between such sequences that expresses that one is a permutation of the other; then

This symmetric function corresponds to the monomial symmetric polynomial mα(X1,…,Xn) for any n largeenough to have the monomial Xα. The distinct monomial symmetric functions are parametrized by the integerpartitions (each mα has a unique representative monomial Xλ with the parts λi in weakly decreasing order).Since any symmetric function containing any of the monomials of some mα must contain all of them with thesame coefficient, each symmetric function can be written as an R-linear combination of monomial symmetricfunctions, and the distinct monomial symmetric functions form a basis of ΛR as R-module.

• The elementary symmetric functions ek, for any natural number k; one has ek = mα where . Asa power series, this is the sum of all distinct products of k distinct indeterminates. This symmetric functioncorresponds to the elementary symmetric polynomial ek(X1,…,Xn) for any n ≥ k.

• The power sum symmetric functions pk, for any positive integer k; one has pk = m(k), the monomial symmetricfunction for the monomial X1

k. This symmetric function corresponds to the power sum symmetric polynomialpk(X1,…,Xn) = X1

k+…+Xnk for any n ≥ 1.

• The complete homogeneous symmetric functions hk, for any natural number k; hk is the sum of all monomial symmetric functions mα where α is a partition of k. As a power series, this is the sum of all monomials of degree k, which is what motivates its name. This symmetric function corresponds to the complete homogeneous

Ring of symmetric functions 84

symmetric polynomial hk(X1,…,Xn) for any n ≥ k.• The Schur functions sλ for any partition λ, which corresponds to the Schur polynomial sλ(X1,…,Xn) for any n

large enough to have the monomial Xλ.There is no power sum symmetric function p0: although it is possible (and in some contexts natural) to define

as a symmetric polynomial in n variables, these values are not compatiblewith the morphisms ρn. The "discriminant" is another example of an expression giving asymmetric polynomial for all n, but not defining any symmetric function. The expressions defining Schurpolynomials as a quotient of alternating polynomials are somewhat similar to that for the discriminant, but thepolynomials sλ(X1,…,Xn) turn out to be compatible for varying n, and therefore do define a symmetric function.

A principle relating symmetric polynomials and symmetric functionsFor any symmetric function P, the corresponding symmetric polynomials in n indeterminates for any natural numbern may be designated by P(X1,…,Xn). The second definition of the ring of symmetric functions implies the followingfundamental principle:

If P and Q are symmetric functions of degree d, then one has the identity of symmetric functions ifand only one has the identity P(X1,…,Xd) = Q(X1,…,Xd) of symmetric polynomials in d indeterminates. In thiscase one has in fact P(X1,…,Xn) = Q(X1,…,Xn) for any number n of indeterminates.

This is because one can always reduce the number of variables by substituting zero for some variables, and one canincrease the number of variables by applying the homomorphisms φn; the definition of those homomorphismsassures that φn(P(X1,…,Xn)) = P(X1,…,Xn+1) (and similarly for Q) whenever n ≥ d. See a proof of Newton'sidentities for an effective application of this principle.

Properties of the ring of symmetric functions

IdentitiesThe ring of symmetric functions is a convenient tool for writing identities between symmetric polynomials that areindependent of the number of indeterminates: in ΛR there is no such number, yet by the above principle any identityin ΛR automatically gives identities the rings of symmetric polynomials over R in any number of indeterminates.Some fundamental identities are

which shows a symmetry between elementary and complete homogeneous symmetric functions; these relations areexplained under complete homogeneous symmetric polynomial.

the Newton identities, which also have a variant for complete homogeneous symmetric functions:

Ring of symmetric functions 85

Structural properties of ΛRImportant properties of ΛR include the following.1. The set of monomial symmetric functions parametrized by partitions form a basis of ΛR as graded R-module,

those parametrized by partitions of d being homogeneous of degree d; the same is true for the set of Schurfunctions (also parametrized by partitions).

2. ΛR is isomorphic as a graded R-algebra to a polynomial ring R[Y1,Y2,…] in infinitely many variables, where Yi isgiven degree i for all i > 0, one isomorphism being the one that sends Yi to ei ∈ ΛR for every i.

3. There is an involutary automorphism ω of ΛR that interchanges the elementary symmetric functions ei and thecomplete homogeneous symmetric function hi for all i. It also sends each power sum symmetric function pi to(−1)i−1 pi, and it permutes the Schur functions among each other, interchanging sλ and sλt where λt is thetranspose partition of λ.

Property 2 is the essence of the fundamental theorem of symmetric polynomials. It immediately implies some otherproperties:• The subring of ΛR generated by its elements of degree at most n is isomorphic to the ring of symmetric

polynomials over R in n variables;• The Hilbert–Poincaré series of ΛR is , the generating function of the integer partitions (this also

follows from property 1);• For every n > 0, the R-module formed by the homogeneous part of ΛR of degree n, modulo its intersection with

the subring generated by its elements of degree strictly less than n, is free of rank 1, and (the image of) en is agenerator of this R-module;

• For every family of symmetric functions (fi)i>0 in which fi is homogeneous of degree i and gives a generator of thefree R-module of the previous point (for all i), there is an alternative isomorphism of graded R-algebras fromR[Y1,Y2,…] as above to ΛR that sends Yi to fi; in other words, the family (fi)i>0 forms a set of free polynomialgenerators of ΛR.

This final point applies in particular to the family (hi)i>0 of complete homogeneous symmetric functions. If Rcontains the field Q of rational numbers, it applies also to the family (pi)i>0 of power sum symmetric functions. Thisexplains why the first n elements of each of these families define sets of symmetric polynomials in n variables thatare free polynomial generators of that ring of symmetric polynomials.The fact that the complete homogeneous symmetric functions form a set of free polynomial generators of ΛR alreadyshows the existence of an automorphism ω sending the elementary symmetric functions to the completehomogeneous ones, as mentioned in property 3. The fact that ω is an involution of ΛR follows from the symmetrybetween elementary and complete homogeneous symmetric functions expressed by the first set of relations givenabove.

Generating functionsThe first definition of ΛR as a subring of R''X''₁,''X''₂,… allows expression the generatingfunctions of several sequences of symmetric functions to be elegantly expressed. Contrary to the relations mentionedearlier, which are internal to ΛR, these expressions involve operations taking place in R[[X1,X2,…;t]] but outside itssubring ΛR[[t]], so they are meaningful only if symmetric functions are viewed as formal power series inindeterminates Xi. We shall write "(X)" after the symmetric functions to stress this interpretation.The generating function for the elementary symmetric functions is

Similarly one has for complete homogeneous symmetric functions

Ring of symmetric functions 86

The obvious fact that explains the symmetry between elementary andcomplete homogeneous symmetric functions. The generating function for the power sum symmetric functions can beexpressed as

((Macdonald, 1979) defines P(t) as Σk>0 pk(X)tk−1, and its expressions therefore lack a factor t with respect to thosegiven here). The two final expressions, involving the formal derivatives of the generating functions E(t) and H(t),imply Newton's identities and their variants for the complete homogeneous symmetric functions. These expressionsare sometimes written as

which amounts to the same, but requires that R contain the rational numbers, so that the logarithm of power serieswith constant term 1 is defined (by ).

References• Macdonald, I. G. Symmetric functions and Hall polynomials. Oxford Mathematical Monographs. The Clarendon

Press, Oxford University Press, Oxford, 1979. viii+180 pp. ISBN 0-19-853530-9 MR84g:05003• Macdonald, I. G. Symmetric functions and Hall polynomials. Second edition. Oxford Mathematical Monographs.

Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995. x+475 pp. ISBN0-19-853489-2 MR96h:05207

• Stanley, Richard P. Enumerative Combinatorics, Vol. 2, Cambridge University Press, 1999. ISBN 0-521-56069-1(hardback) ISBN 0-521-78987-7 (paperback).

Schur polynomial 87

Schur polynomialIn mathematics, Schur polynomials, named after Issai Schur, are certain symmetric polynomials in n variables,indexed by partitions, that generalize the elementary symmetric polynomials and the complete homogeneoussymmetric polynomials. In representation theory they are the characters of irreducible representations of the generallinear groups. The Schur polynomials form a linear basis for the space of all symmetric polynomials. Any product ofSchur functions can be written as a linear combination of Schur polynomials with non-negative integral coefficients;the values of these coefficients is given combinatorially by the Littlewood-Richardson rule. More generally, skewSchur polynomials are associated with pairs of partitions and have similar properties to Schur polynomials.

DefinitionSchur polynomials correspond to integer partitions. Given a partition

(where each is a non-negative integer), the following functions are alternating polynomials (in other words theychange sign under any transposition of the variables):

Since they are alternating, they are all divisible by the Vandermonde determinant:

The Schur polynomials are defined as the ratio:

This is a symmetric function because the numerator and denominator are both alternating, and a polynomial since allalternating polynomials are divisible by the Vandermonde determinant.

PropertiesThe degree d Schur polynomials in n variables are a linear basis for the space of homogeneous degree d symmetricpolynomials in n variables.The first Giambelli formula gives explicit expression of Schur polynomials as a polynomial in the completehomogeneous symmetric polynomials:

The second Giambelli formula gives explicit expression of Schur polynomials as polynomials in the elementarysymmetric polynomials:

where is a dual partition to

Schur polynomial 88

These two formulas are also known as "determinantal formulas" and the first one is known as the Jacobi-Trudyidentity.For a partition , the Schur function is a sum of monomials:

where the summation is over all semistandard Young tableaux of shape ; the exponents give theweight of , in other words each counts the occurrences of the number in . This can be shown to beequivalent to the definition from the first Giambelli formula using the Lindström–Gessel–Viennot lemma (asoutlined on that page).Schur polynomials sλ can be expressed as linear combinations of monomial symmetric functions mμ withnon-negative integer coefficients Kλμ called Kostka numbers:

ExampleThe following extended example should help clarify these ideas. Consider the case n = 3, d = 4. Using Ferrersdiagrams or some other method, we find that there are just four partitions of 4 into at most three parts. We have

and so forth. Summarizing:1.2.3.4.Every homogeneous degree-four symmetric polynomial in three variables can be expressed as a unique linearcombination of these four Schur polynomials, and this combination can again be found using a Gröbner basis for anappropriate elimination order. For example,

is obviously a symmetric polynomial which is homogeneous of degree four, and we have

Schur polynomial 89

Relation to representation theoryThe Schur polynomials occur in the representation theory of the symmetric groups, general linear groups, andunitary groups, and in fact this is how they arose. The Weyl character formula implies that the Schur polynomials arethe characters of finite dimensional irreducible representations of the general linear groups, and helps to generalizeSchur's work to other compact and semisimple Lie groups.Several expressions arise for this relation, one of the most important being the expansion of the Schur functions in terms of the symmetric power functions . If we write for the character of the representation of

the symmetric group indexed by the partition evaluated at elements of cycle type indexed by the partition ,then

where means that the partition has parts of length .

Skew Schur functionsSkew Schur functions sλ/μ depend on two partitions λ and μ, and can be defined by the property

References• Macdonald, I. G. (1995). Symmetric functions and Hall polynomials [1]. Oxford Mathematical Monographs (2nd

ed.). The Clarendon Press Oxford University Press. ISBN 978-0-19-853489-1. MR1354144• Sagan, Bruce E. (2001), "Schur functions in algebraic combinatorics" [1], in Hazewinkel, Michiel, Encyclopedia

of Mathematics, Springer, ISBN 978-1556080104• Sturmfels, Bernd (1993). Algorithms in Invariant Theory. New York: Springer. ISBN 0-387-82445-6.

References[1] http:/ / www. encyclopediaofmath. org/ index. php?title=s/ s120040

Symbolic method 90

Symbolic methodIn mathematics, the symbolic method in invariant theory is an algorithm developed by Arthur Cayley, SiegfriedHeinrich Aronhold (1858), Alfred Clebsch (1861), and Paul Gordan (1887) in the 19th century for computinginvariants of algebraic forms. It is based on treating the form as if it were a power of a degree one form.

Symbolic notationThe symbolic method uses a compact but rather confusing and mysterious notation for invariants, depending on theintroduction of new symbols a, b, c, ... (from which the symbolic method gets its name) with apparentlycontradictory properties.

Example: the discriminant of a binary quadratic formThese symbols can be explained by the following example from (Gordan 1887, volume 2, pages 1-3). Suppose that

is a binary quadratic form with an invariant given by the discriminant

The symbolic representation of the discriminant is

where a and b are the symbols. The meaning of the expression (ab)2 is as follows. First of all, (ab) is a shorthandform for the determinant of a matrix whose rows are a1, a2 and b1, b2, so

Squaring this we get

Next we pretend that

so that

and we ignore the fact that this does not seem to make sense if f is not a power of a linear form. Substituting thesevalues gives

Higher degreesMore generally if

is a binary form of higher degree, then one introduces new variables a1, a2, b1, b2, c1, c2, with the properties

What this means is that the following two vector spaces are naturally isomorphic:• The vector space of homogeneous polynomials in A0,...An of degree m• The vector space of polynomials in 2m variables a1, a2, b1, b2, c1, c2, ... that have degree n in each of the m pairs

of variables (a1, a2), (b1, b2), (c1, c2), ... and are symmetric under permutations of the m symbols a, b, ....,

Symbolic method 91

The isomorphism is given by mapping aa , bb , .... to Aj. This mapping does not preserve products of

polynomials.

More variablesThe extension to a form f in more than two variables x1, x2,x3,... is similar: one introduces symbols a1, a2,a3 and soon with the properties

References• Aronhold, Siegfried Heinrich (1858), "Theorie der homogenen Functionen dritten Grades von drei

Veränderlichen." [1] (in German), Journal für die reine und angewandte Mathematik 55: 97–191,ISSN 0075-4102

• Clebsch, A. (1861), "Ueber symbolische Darstellung algebraischer Formen" [2] (in German), Journal für Reineund Angewandte Mathematik 59: 1–62, ISSN 0075-4102

• Dieudonné, Jean; Carrell, James B. (1970), "Invariant theory, old and new" [3], Advances in Mathematics 4: 1–80,pages 32–37, "Invariants of n-ary forms: the symbolic method. Reprinted as Dieudonné, Jean; Carrell, James B.(1971), Invariant theory, old and new, Academic Press, ISBN 0122155408

• Dolgachev, Igor (2003), Lectures on invariant theory, London Mathematical Society Lecture Note Series, 296,Cambridge University Press, doi:10.1017/CBO9780511615436, ISBN 978-0-521-52548-0, MR2004511

• Gordan, Paul (1887), Kerschensteiner, Georg, ed., Vorlesungen über Invariantentheorie [4] (2nd ed.), New York:Chelsea Publishing Co., ISBN 978-0-8284-0328-3, MR917266

• Grace, John Hilton; Young, Alfred (1903), The Algebra of invariants, Cambridge University Press• Hilbert, David (1993) [1897], Theory of algebraic invariants [1], Cambridge University Press,

ISBN 978-0-521-44457-6, MR1266168• Koh, Sebastian S., ed. (1987), Invariant Theory, Lecture Notes in Mathematics, 1278, ISBN 3540183604• Kung, Joseph P. S.; Rota, Gian-Carlo (1984), "The invariant theory of binary forms" [1], American Mathematical

Society. Bulletin. New Series 10 (1): 27–85, doi:10.1090/S0273-0979-1984-15188-7, ISSN 0002-9904,MR722856

References[1] http:/ / resolver. sub. uni-goettingen. de/ purl?GDZPPN00215028X[2] http:/ / resolver. sub. uni-goettingen. de/ purl?PPN243919689_0059[3] http:/ / www. sciencedirect. com/ science?_ob=ArticleURL& _udi=B6W9F-4D7JKM7-1& _user=1495569&

_coverDate=02%2F28%2F1970& _rdoc=1& _fmt=high& _orig=search& _origin=search& _sort=d& _docanchor=& view=c&_acct=C000053194& _version=1& _urlVersion=0& _userid=1495569& md5=66afdc20312efc67ffb295764befef82& searchtype=a

[4] http:/ / books. google. com/ books?isbn=978-0-8284-0328-3

Trace identity 92

Trace identityIn mathematics, a trace identity is any equation involving the trace of a matrix. For example, the Cayley–Hamiltontheorem says that every matrix satisfies its own characteristic polynomial.Trace identities are invariant under simultaneous conjugation. They are frequently used in the invariant theory of n×nmatrices to find the generators and relations of the ring of invariants, and therefore are useful in answering questionssimilar to that posed by Hilbert's fourteenth problem.

Examples• By the Cayley–Hamilton theorem, all matrices satisfy

• All matrices satisfy

TransvectantIn mathematical invariant theory, a transvectant is an invariant formed from n invariants in n variables usingCayley's omega process.

DefinitionIf Q1,...,Qn are functions of n variables x = (x1,...,xn) and r ≥ 0 is an integer then the rth transvectant of thesefunctions is a function of n variables given by

where Ω is Cayley's omega process, the tensor product means take a product of functions with different variablesx1,..., xn, and tr means set all the vectors xk equal.

ExamplesThe zeroth transvectant is the product of the n functions.The first transvectant is the Jacobian determinant of the n functions.The second transvectant is a constant times the completely polarized form of the Hessian of the n functions.

References• Olver, Peter J. (1999), Classical invariant theory, Cambridge University Press, ISBN 978-0-521-55821-1• Olver, Peter J.; Sanders, Jan A. (2000), "Transvectants, modular forms, and the Heisenberg algebra", Advances in

Applied Mathematics 25 (3): 252–283, doi:10.1006/aama.2000.0700, ISSN 0196-8858, MR1783553

Article Sources and Contributors 93

Article Sources and ContributorsInvariant theory  Source: http://en.wikipedia.org/w/index.php?oldid=446056664  Contributors: Arcfrk, Charles Matthews, Darij, David Eppstein, Giftlite, Headbomb, Hillman, Javanbakht,Ligulem, Maxdlink, Michael Larsen, Nbarth, Omnipaedista, PV=nRT, Polyade, R.e.b., Rgdboer, Rvollmert, Sherbrooke, The enemies of god, Vegasprof, 8 anonymous edits

Bracket algebra  Source: http://en.wikipedia.org/w/index.php?oldid=424246677  Contributors: Cronholm144, Geometry guy, Michael Hardy, Michael Slone, Nilradical, Rjwilmsi

Capelli's identity  Source: http://en.wikipedia.org/w/index.php?oldid=465443809  Contributors: Alexander Chervov, Charles Matthews, Darij, David Eppstein, Giftlite, Headbomb, MichaelHardy, Mild Bill Hiccup, R.e.b., Rjwilmsi, Woohookitty, 11 anonymous edits

Catalecticant  Source: http://en.wikipedia.org/w/index.php?oldid=446768104  Contributors: Headbomb, Michael Hardy, R.e.b., Rjwilmsi

Cayley's Ω process  Source: http://en.wikipedia.org/w/index.php?oldid=446768222  Contributors: Alexander Chervov, Anthony Appleyard, Headbomb, JackSchmidt, Michael Hardy, R.e.b., 1anonymous edits

Chevalley–Shephard–Todd theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455616003  Contributors: Arcfrk, Closedmouth, Geometry guy, Giftlite, Headbomb, Michael Hardy,R.e.b., Rjwilmsi, 4 anonymous edits

The Classical Groups  Source: http://en.wikipedia.org/w/index.php?oldid=457259483  Contributors: Epbr123, Headbomb, JIP, Michael Hardy, R.e.b., 1 anonymous edits

Differential invariant  Source: http://en.wikipedia.org/w/index.php?oldid=411055535  Contributors: Charvest, Henry Delforn, Michael Hardy, Sławomir Biały

Geometric invariant theory  Source: http://en.wikipedia.org/w/index.php?oldid=421280118  Contributors: Altenmann, Arcfrk, Arthur Rubin, Basemaze, Charles Matthews, Eigenlambda,Gauge, Giftlite, Hillman, JCSantos, JackSchmidt, Paul August, R.e.b., RobHar, RobinK, Stca74, 7 anonymous edits

Gröbner basis  Source: http://en.wikipedia.org/w/index.php?oldid=458311362  Contributors: (:Julien:), 4pq1injbok, Allansteel, Archimerged, Arthur Rubin, AugPi, AxelBoldt,Baccala@freesoft.org, Betacommand, Bricken, CRGreathouse, Charles Matthews, Chris the speller, D.Lazard, DR2006kl, Darij, FlyHigh, Gauge, Gene Nygaard, Giftlite, Hannes Eder, Hillman,Hqb, Jaraalbe, JonMcLoone, Kevinatilusa, Lambiam, Liberatus, Liberio, LutzL, M.Moreno-Maza, Maxal, Michael Hardy, Miketeri, Monzloff, NBeale, Nbarth, Ntmatter, NymphadoraTonks,Oleg Alexandrov, Orangutan123, Pascal.Tesson, Paul August, Populus, Rodhullandemu, Ruud Koot, Shreevatsa, Simplex, Sirius533, Spoon!, T.huckstep, T68492, Theodyl, Turgidson,WATARU, Williamaffleck, Ylloh, Zadneram, 63 anonymous edits

Haboush's theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455765461  Contributors: BeteNoir, Charles Matthews, David Eppstein, DavidCBryant, Giftlite, Jeff3000, R'n'B, R.e.b.,Ringspectrum, Rjwilmsi, RobHar, Sodin, 4 anonymous edits

Hall algebra  Source: http://en.wikipedia.org/w/index.php?oldid=435595563  Contributors: Arcfrk, Charvest, Giftlite, Headbomb, Jitse Niesen, Michael Hardy, Omnipaedista, R.e.b., Turgidson,Vanish2

Hilbert's basis theorem  Source: http://en.wikipedia.org/w/index.php?oldid=462451074  Contributors: Ad4m, Arcfrk, AxelBoldt, BeteNoir, Brockert, Bryan Derksen, Charles Matthews,Conversion script, Drusus 0, Gaius Cornelius, Giftlite, Guardian of Light, Hillman, ICPalm, Jowa fan, Jxr, Kinu, LilHelpa, MathMartin, Michael Slone, Oleg Alexandrov, R.e.b., Randomblue,Sodin, Tobias Bergemann, Vivacissamamente, Waltpohl, 19 anonymous edits

Hilbert's fourteenth problem  Source: http://en.wikipedia.org/w/index.php?oldid=453903680  Contributors: Arcfrk, BertSeghers, Charles Matthews, GregorB, Gro-Tsen, Headbomb,Lejean2000, Mattbuck, Michael Hardy, Natalya, Polyade, R.e.b., RetiredUser2, Rjwilmsi, Sango123, Silverfish, Zundark, 1 anonymous edits

Hilbert's syzygy theorem  Source: http://en.wikipedia.org/w/index.php?oldid=455609953  Contributors: Arcfrk, BeteNoir, Charles Matthews, Giftlite, Grendelkhan, Myrizio, Nick Number,Sodin, Ylloh, Zaphod Beeblebrox, 12 anonymous edits

Hodge bundle  Source: http://en.wikipedia.org/w/index.php?oldid=429915100  Contributors: CBM, Charles Matthews, DESiegel, Michael Hardy, Mono, RobHar, Zvonkine

Invariant estimator  Source: http://en.wikipedia.org/w/index.php?oldid=447607640  Contributors: 23dp, 3mta3, Backtable, Headbomb, Iridescent, Jack-A-Roe, Kiefer.Wolfowitz, Melcombe,Rajagiryes, The Utahraptor, Zvika, 4 anonymous edits

Invariant of a binary form  Source: http://en.wikipedia.org/w/index.php?oldid=435578719  Contributors: Charles Matthews, Michael Hardy, R.e.b., Rjwilmsi, 2 anonymous edits

Invariant polynomial  Source: http://en.wikipedia.org/w/index.php?oldid=272074880  Contributors: Charles Matthews, Dialectric, Fredrik, Michael Hardy, Nabla, Silverfish

Invariants of tensors  Source: http://en.wikipedia.org/w/index.php?oldid=450323638  Contributors: Alex Bakharev, Ana109, Ariel.amir, BenFrantzDale, Cerniagigante, Charles Matthews,Ebarbero, Gcranston, Hillman, JRSpriggs, Konrad West, Michael Hardy, Mpatel, Pearle, Quiddity, RDT2, Rbdupaix, Sverdrup, TimBentley, Tomeasy, WhiteHatLurker, 13 anonymous edits

Kostant polynomial  Source: http://en.wikipedia.org/w/index.php?oldid=456932655  Contributors: Chronulator, David Eppstein, EdoDodo, Giftlite, Makotoy, Mathsci, Michael Hardy, 1anonymous edits

Littlewood–Richardson rule  Source: http://en.wikipedia.org/w/index.php?oldid=456456576  Contributors: Ashton1983, Charles Matthews, Giftlite, JackSchmidt, Jambaugh, KarlJacobi,KathrynLybarger, Magister Mathematicae, Marc van Leeuwen, Mhym, Michael Hardy, Plrk, R.e.b., Rjwilmsi, Shreevatsa, WikHead, Zdaugherty, 3 anonymous edits

Modular invariant of a group  Source: http://en.wikipedia.org/w/index.php?oldid=447948593  Contributors: Headbomb, Michael Hardy, R.e.b., 1 anonymous edits

Moduli space  Source: http://en.wikipedia.org/w/index.php?oldid=451968066  Contributors: 7&6=thirteen, AmarChandra, AxelBoldt, C quest000, CRGreathouse, Charles Matthews,Dreamseeker02139, Edward, False vacuum, Gauge, Giftlite, Hesam7, Hillman, Jakob.scholbach, JesseW, K igor k, Keenan Pepper, Lumidek, Masnevets, Michael Hardy, Michael Larsen,Michael Slone, Mild Bill Hiccup, Nbarth, Oleg Alexandrov, RDBury, Rvollmert, Stca74, Stephan Spahn, Tobias Bergemann, Tong, Vanish2, 79 anonymous edits

Molien series  Source: http://en.wikipedia.org/w/index.php?oldid=447948785  Contributors: Charles Matthews, Darij, Giftlite, Headbomb, JackSchmidt, Michael Slone, RobinK, Simplifix

Newton's identities  Source: http://en.wikipedia.org/w/index.php?oldid=461842533  Contributors: Andreasmperu, Charles Matthews, Chuunen Baka, DA3N, David Eppstein, Dogaroon,Edinborgarstefan, Gauge, Giftlite, Headbomb, Hillman, Icairns, JYOuyang, JerroldPease-Atlanta, KSmrq, Kbdank71, KlappCK, Ligulem, LutzL, Marc van Leeuwen, Mhym, Michael Hardy,Mild Bill Hiccup, Mulanhua, Paul August, Plasticup, Quantling, Qutezuce, Razorflame, Reinyday, Rich Farmbrough, Salgueiro, Shlomi Hillel, Simon12, T0, That Guy, From That Show!, TonySidaway, Zaslav, 27 anonymous edits

Polynomial ring  Source: http://en.wikipedia.org/w/index.php?oldid=453033291  Contributors: Ahoerstemeier, Aiden Fisher, Alephcero, Algebran, Anonymous Dissident, Arcfrk, Arthur Rubin,Bo Jacoby, Cacadril, Calle, Cflm001, Charles Matthews, Classicalecon, D.Lazard, D.M. from Ukraine, DonDiego, Flyhighplato, Fropuff, Gauss, Giftlite, Gwaihir, Henry Delforn, Hillman,Ht686rg90, Hyginsberg, JackSchmidt, Justin W Smith, KnowledgeOfSelf, Linas, MFH, Marc van Leeuwen, MathMartin, Mathaxiom, Michael Hardy, Mmernex, Oleg Alexandrov, Point-settopologist, Rgdboer, Rjwilmsi, RobHar, Salix alba, Silly rabbit, Solvecolorer, Taxman, TheObtuseAngleOfDoom, WATARU, Waggers, Waltpohl, Zero0000, 饒彥偉, 30 anonymous edits

Quantum invariant  Source: http://en.wikipedia.org/w/index.php?oldid=389312962  Contributors: Bender235, David Eppstein, Henry Delforn, Ilmari Karonen, Michael Hardy, Moonriddengirl,NuclearWarfare

Radical polynomial  Source: http://en.wikipedia.org/w/index.php?oldid=221034708  Contributors: Charles Matthews, Geometry guy, Michael Hardy, Vanish2, Vipul, 2 anonymous edits

Reynolds operator  Source: http://en.wikipedia.org/w/index.php?oldid=447967719  Contributors: Charles Matthews, Charlesreid1, Crowsnest, Headbomb, Lohengrin9, Michael Hardy, R.e.b.,WikHead

Riemann invariant  Source: http://en.wikipedia.org/w/index.php?oldid=461297761  Contributors: Headbomb, Richmarc5, Sonia, 3 anonymous edits

Ring of symmetric functions  Source: http://en.wikipedia.org/w/index.php?oldid=457431167  Contributors: Ahoerstemeier, Arthur Rubin, Basemaze, BenFrantzDale, CBM, Charles Matthews, Darij, Giftlite, Haonhien, Hillman, John of Reading, Marc van Leeuwen, Mhym, Michael Hardy, Nbarth, Oleg Alexandrov, PV=nRT, Patrick, Sbilley, Tabletop, Woohookitty, Zaslav, Петър

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Петров, 10 anonymous edits

Schur polynomial  Source: http://en.wikipedia.org/w/index.php?oldid=447504675  Contributors: Charles Matthews, Col tom, Darij, Gene496, Giftlite, Hillman, JackSchmidt, Ligulem, Marc vanLeeuwen, Michael Hardy, Nbarth, Plamenkoev, R.e.b., Sam Derbyshire, Skinnerd, Tcamps42, That Guy, From That Show!, Vegasprof, W9q, Zaslav, 20 anonymous edits

Symbolic method  Source: http://en.wikipedia.org/w/index.php?oldid=460799767  Contributors: Charles Matthews, Doetoe, Headbomb, Hillman, IronGargoyle, Jrdioko, MONGO, Mathsci,Mets501, Oleg Alexandrov, R.e.b., R.e.s., Tango, 16 anonymous edits

Trace identity  Source: http://en.wikipedia.org/w/index.php?oldid=404991899  Contributors: Algebraist, Alksentrs, Michael Hardy, Nono64, R'n'B, Tobias Bergemann, Triathematician, 2anonymous edits

Transvectant  Source: http://en.wikipedia.org/w/index.php?oldid=447977810  Contributors: Headbomb, Michael Hardy, R.e.b.

Image Sources, Licenses and Contributors 95

Image Sources, Licenses and ContributorsImage:LR tableau of shape (4,3,2)-(2,1) word 112123.svg  Source: http://en.wikipedia.org/w/index.php?title=File:LR_tableau_of_shape_(4,3,2)-(2,1)_word_112123.svg  License: CreativeCommons Attribution-Sharealike 3.0  Contributors: Marc van Leeuwen (talk)Image:LR tableau of shape (4,3,2)-(2,1) word 112213.svg  Source: http://en.wikipedia.org/w/index.php?title=File:LR_tableau_of_shape_(4,3,2)-(2,1)_word_112213.svg  License: CreativeCommons Attribution-Sharealike 3.0  Contributors: Marc van Leeuwen (talk)File:Schur functions rectangular example.png  Source: http://en.wikipedia.org/w/index.php?title=File:Schur_functions_rectangular_example.png  License: Creative CommonsAttribution-Sharealike 3.0  Contributors: -

License 96

LicenseCreative Commons Attribution-Share Alike 3.0 Unported//creativecommons.org/licenses/by-sa/3.0/