Stable Representation Theory - Mathematical Sciencesmath.iupui.edu/~dramras/UTEP.pdf · Cohomology...

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Stable Representation Theory Daniel A. Ramras Department of Mathematical Sciences New Mexico State University September 10, 2010 Daniel A. Ramras Stable Representation Theory

Transcript of Stable Representation Theory - Mathematical Sciencesmath.iupui.edu/~dramras/UTEP.pdf · Cohomology...

Page 1: Stable Representation Theory - Mathematical Sciencesmath.iupui.edu/~dramras/UTEP.pdf · Cohomology and Classifying Spaces The cohomology groups Hk (G) can be defined Algebraically:

Stable Representation Theory

Daniel A. RamrasDepartment of Mathematical Sciences

New Mexico State University

September 10, 2010

Daniel A. Ramras Stable Representation Theory

Page 2: Stable Representation Theory - Mathematical Sciencesmath.iupui.edu/~dramras/UTEP.pdf · Cohomology and Classifying Spaces The cohomology groups Hk (G) can be defined Algebraically:

Representation Theory

Given a (finitely generated, discrete group) G, we want todescribe the homomorphisms

! : G !" U(n)

and understand what these representations say about G.

Can one find an algebraic or combinatorial description of allrepresentations?

Difficult or impossible in general.

Today: Topological approachUsing the natural topology of U(n) # Mn!n(C), we’ll considerthe space Hom(G, U(n)) of representations.

Daniel A. Ramras Stable Representation Theory

Page 3: Stable Representation Theory - Mathematical Sciencesmath.iupui.edu/~dramras/UTEP.pdf · Cohomology and Classifying Spaces The cohomology groups Hk (G) can be defined Algebraically:

Spaces of Representations

If G is generated by k elements, then

Hom(G, U(n)) ! U(n)k .

We’ll also consider isomorphism classes of representations:

! "= A!A!1

for any A # U(n).

The space of isomorphism classes is then

Hom(G, U(n))/U(n) = Hom(G, U(n))/(! " A!A!1)

Daniel A. Ramras Stable Representation Theory

Page 4: Stable Representation Theory - Mathematical Sciencesmath.iupui.edu/~dramras/UTEP.pdf · Cohomology and Classifying Spaces The cohomology groups Hk (G) can be defined Algebraically:

Example: Finite Groups

If G is finite, then a representation ! : G! U(n) is determinedup to isomorphism by its character,

Trace(!) : G! C

The traceis a continuous invarianttakes on only finitely many values.

Conclusion: Hom(G, U(n))/U(n) is a finite, discrete space!

However, there is interesting topology lurking here...

Daniel A. Ramras Stable Representation Theory

Page 5: Stable Representation Theory - Mathematical Sciencesmath.iupui.edu/~dramras/UTEP.pdf · Cohomology and Classifying Spaces The cohomology groups Hk (G) can be defined Algebraically:

The Atiyah–Hirzebruch–Segal Theorem

The Representation Ring:

R(G) = {[!]! ["] : !, " are unitary representations of G}

Consists of formal differences between isomorphismclasses of representationsBlock sum, tensor product give addition and multiplication.

Theorem (A-H-S)For any finite group G, there is a spectral sequence

H!(G) =" R(G).

Algebraic procedure for constructing R(G) out of thecohomology of G.

Daniel A. Ramras Stable Representation Theory

Page 6: Stable Representation Theory - Mathematical Sciencesmath.iupui.edu/~dramras/UTEP.pdf · Cohomology and Classifying Spaces The cohomology groups Hk (G) can be defined Algebraically:

Cohomology and Classifying Spaces

The cohomology groups Hk (G) can be definedAlgebraically: Hochschild cohomologyTopologically: for a space X , the group Hk (X ) measuresk–dimensional “combinatorial holes” in X .

DefinitionA classifying space for a group G is a quotient space X/G,where X is a contractible simplicial complex on which G actsfreely.

FactAll classifying spaces for G have the same cohomology,denoted H!(G).

Daniel A. Ramras Stable Representation Theory

Page 7: Stable Representation Theory - Mathematical Sciencesmath.iupui.edu/~dramras/UTEP.pdf · Cohomology and Classifying Spaces The cohomology groups Hk (G) can be defined Algebraically:

Representations and Cohomology

Lots of mathematics has grown out of attempts to betterunderstand the Atiyah–Hirzebruch–Segal Theorem.

Geometric Rigidity: Are all classifying spaces for a givengroup G homeomorphic to one another?

Borel Conjecture:YES, if these classifying spaces are high-dimensionalmanifolds (of the same dimension).

Studied via obstruction theory, and depends on relatingmodules over Z[G] (“representations”) to a modifiedversion of the homology of G.

Daniel A. Ramras Stable Representation Theory

Page 8: Stable Representation Theory - Mathematical Sciencesmath.iupui.edu/~dramras/UTEP.pdf · Cohomology and Classifying Spaces The cohomology groups Hk (G) can be defined Algebraically:

Representations and Cohomology

Novikov Conjecture: asks whether certain differentialinvariants of manifolds are actually homotopy invariant.

Studied by relating certain Banach algebras built from G(“representations”) to another modified homology theory.

Quillen–Lichtenbaum Conjecture: relates algebraicK –theory to Galois cohomology theory.

Today:We’ll examine representations and cohomology of infinite,discrete groups, where the connection is much more explicitand geometric.

Goal: Shed light on some of these more abstract areas.

Daniel A. Ramras Stable Representation Theory

Page 9: Stable Representation Theory - Mathematical Sciencesmath.iupui.edu/~dramras/UTEP.pdf · Cohomology and Classifying Spaces The cohomology groups Hk (G) can be defined Algebraically:

Representations and cohomology of infinite discretegroups

Example (G = Z)Hom(Z, U(n)) = U(n)

Hom(Z, U(n))/U(n) = U(n)/(A ! PAP!1)

Spectral Theorem:Every A " U(n) has n eigenvalues, when counted withmultiplicity.

This yields an eigenvalue map

Hom(Z, U(n))/U(n) #$ (S1 % · · ·% S1! "# $

n

)/!n =: Symn(S1).

This map is a homeomorphism by Rouche’s Theorem.

Daniel A. Ramras Stable Representation Theory

Page 10: Stable Representation Theory - Mathematical Sciencesmath.iupui.edu/~dramras/UTEP.pdf · Cohomology and Classifying Spaces The cohomology groups Hk (G) can be defined Algebraically:

Stabilization

Something interesting happens when we stabilize: let n !":!

nHom(Z, U(n))/U(n) #=

!

nSymn(S1) = Sym!(S1)

Theorem (Dold–Thom)For any (based) simplicial complex X, the homotopy groups ofSym!(X ) are isomorphic to the homology groups of X .

The homotopy groups !k (Z ) measure k–dimensional“spherical holes” in Z :

!k (Z ) = {f : Sk $! Z}/homotopy.

Conclusion: !"Hom(Z, U)/U #= H"(S1) = H"(Z),S1 is a classifying space for Z, because R/Z = S1.

Daniel A. Ramras Stable Representation Theory

Page 11: Stable Representation Theory - Mathematical Sciencesmath.iupui.edu/~dramras/UTEP.pdf · Cohomology and Classifying Spaces The cohomology groups Hk (G) can be defined Algebraically:

Stable representation theory of Zn

By similar methods, we can analyze the free abelian group Zk :

Rk/Zk = (R/Z)k = (S1)k , so the k–dimensional torus is aclassifying space for Zk .Commuting unitary matrices are simultaneouslydiagonalizable, so to every representation of Zk we canassociate its eigenvalues:

Hom(Zk , U(n))/U(n) != Symn((S1)k ).

Once again, the Dold–Thom Theorem tells us that

!!Hom(Zk , U)/U != H!(Zk ).

Daniel A. Ramras Stable Representation Theory

Page 12: Stable Representation Theory - Mathematical Sciencesmath.iupui.edu/~dramras/UTEP.pdf · Cohomology and Classifying Spaces The cohomology groups Hk (G) can be defined Algebraically:

Surface Groups

Any Riemann surface S is a classifying space for itsfundamental group: if S has genus g,

!1S = !a1, b1, . . . , ag , bg |[a1, b1] · · · [ag , bg] = 1".

Theorem (R.)For each Riemann surface S, there is a homotopy equivalenceHom(!1S, U)/U # Sym!(S), and hence

!k Hom(!1S, U)/U $= Hk (!1S).

Relies on Morse Theory for the Yang–Mills functional andstable homotopy theory.

Heuristic: Representations ! Vector bundles over S !

Characteristic (co)homology classes

Daniel A. Ramras Stable Representation Theory

Page 13: Stable Representation Theory - Mathematical Sciencesmath.iupui.edu/~dramras/UTEP.pdf · Cohomology and Classifying Spaces The cohomology groups Hk (G) can be defined Algebraically:

Crystallographic Groups

DefinitionA crystallographic group is a discrete subgroup! < Isom(Rk ) = Rk ! O(k), such that Rk/! is compact.

FactIf ! is a torsion-free crystallographic group, then ! acts freely onRk , and Rk/! is a classifying space for !.

The translations in ! form a free abelian subgroup T != Zk

of finite index.

Consequence: The irreducible representations of ! havedimension at most |!/T |, making computations feasible.

Daniel A. Ramras Stable Representation Theory

Page 14: Stable Representation Theory - Mathematical Sciencesmath.iupui.edu/~dramras/UTEP.pdf · Cohomology and Classifying Spaces The cohomology groups Hk (G) can be defined Algebraically:

A Crystallographic Family

Example

The group !k = !t1, . . . , tk , a | [ti , tj ] = 1, atia!1 = t!1i " is

crystallographic.

T = !t1, . . . , tk , a2" #= Zk+1,!k/T = !a"/!a2" #= Z/2Z.

!k acts on Rk+1 by the isometries

v ti$% v + ei

v a$%

!

"""#

&1. . .

&11

$

%%%&v +

12

ek+1

Daniel A. Ramras Stable Representation Theory

Page 15: Stable Representation Theory - Mathematical Sciencesmath.iupui.edu/~dramras/UTEP.pdf · Cohomology and Classifying Spaces The cohomology groups Hk (G) can be defined Algebraically:

The classifying space

The classifying space Rk+1/!k has the form

(Rk+1/Zk+1)/ (Z/2Z)

= (S1)k+1/(z1, . . . , zk , a) ! (z!11 , . . . , z!1

k ,"a).

Example (The Klein Bottle)When k = 1, this quotient space is the usual model for theKlein Bottle!

Daniel A. Ramras Stable Representation Theory

Page 16: Stable Representation Theory - Mathematical Sciencesmath.iupui.edu/~dramras/UTEP.pdf · Cohomology and Classifying Spaces The cohomology groups Hk (G) can be defined Algebraically:

The irreducible representations of !k

Irreducible representations of !k have dimension ! 2.

Recall: ! is irreducible if ! !!

" 00 #

".

We will analyze the closure of the 2–dimensionalirreducible representations.

Fact

Irr2(!k ) =

#$!#i 00 #!1

i

",

!0 $1 0

"%: #i , $ " S1

&/ #,

where the equivalence relation # is given by:$!

#i 00 #!1

i

",

!0 $1 0

"%#

$!#!1

i 00 #i

",

!0 $1 0

"%.

Daniel A. Ramras Stable Representation Theory

Page 17: Stable Representation Theory - Mathematical Sciencesmath.iupui.edu/~dramras/UTEP.pdf · Cohomology and Classifying Spaces The cohomology groups Hk (G) can be defined Algebraically:

Comparing Representations and Cohomology

Irr2(!k ) = (S1)k+1/ (Z/2Z), with action

(!1, . . . , !k , ") !" (!!11 , . . . , !!1

k , ").

We saw that Rk+1/!k = (S1)k+1/ (Z/2Z), with action

(z1, . . . , zk , a) # (z!11 , . . . , z!1

k ,$a).

The actions are homotopic to one another, and the cohomologygroups of the quotients have the same rank (Grothendieck).

Theorem (R.)

For each group !k = %t1, . . . , tk , a | [ti , tj ] = 1, atia!1 = t!1i &,

Rank (#qHom(!k , U)/U) = Rank Hq(!k )

(except possibly when q = 1, 2).

Daniel A. Ramras Stable Representation Theory

Page 18: Stable Representation Theory - Mathematical Sciencesmath.iupui.edu/~dramras/UTEP.pdf · Cohomology and Classifying Spaces The cohomology groups Hk (G) can be defined Algebraically:

Stabilization versus Group Completion

In order to extend these ideas to all crystallographic groups, weneed to analyze the idea of stabilization.

In the A–H–S Theorem, we formed R(G) by adding formalinverses to each representation of G:

This is called Group Completion.

For infinite discrete groups, we stabilized by letting thedimensions of the representations tend to infinity.

Question:How can we relate group completion and stabilization?

ExampleThe integers Z can be built from the natural numbers N byeither process.

Daniel A. Ramras Stable Representation Theory

Page 19: Stable Representation Theory - Mathematical Sciencesmath.iupui.edu/~dramras/UTEP.pdf · Cohomology and Classifying Spaces The cohomology groups Hk (G) can be defined Algebraically:

The Grothendieck Group and Stabilization

The group completion, or Grothendieck Group, of a (niceenough) abelian topological monoid A is the group

Gr(A) = A! A/ "

where (a1, a2) " (a!1, a!2) if a1 + a!2 + a = a!1 + a2 + a for some a.Analogous to the construction of fraction fields.

PropositionIf A is a discrete abelian monoid and a0 # A satisfies:

$a # A, % a # A such that a + a # &a0',

thenGr(A) "= lim

"(A +a0() A +a0() A +a0() · · · ).

Example: every vector bundle V ) X has an orthogonalcomplement: V * V# "= X ! RN # &X ! R'.

Daniel A. Ramras Stable Representation Theory

Page 20: Stable Representation Theory - Mathematical Sciencesmath.iupui.edu/~dramras/UTEP.pdf · Cohomology and Classifying Spaces The cohomology groups Hk (G) can be defined Algebraically:

Group completion versus stabilization forrepresentation spaces

Definition

The deformation representation ring Rdef(G) is the groupcompletion of

!n Hom(G, U(n))/U(n).

The groups we’ve focused on today satisfy:

For all ! : G ! U(n), there exists k such that

!" !k # I,

where I is the trivial representation.

This implies that for all these groups G,

Rdef(G) # lim!1!"

"#

nHom(G, U(n))/U(n)

$

Daniel A. Ramras Stable Representation Theory

Page 21: Stable Representation Theory - Mathematical Sciencesmath.iupui.edu/~dramras/UTEP.pdf · Cohomology and Classifying Spaces The cohomology groups Hk (G) can be defined Algebraically:

General results for crystallographic groups

Theorem (R.)

For any crystallographic group ! < Isom(Rk ), the homotopygroups of the deformation representation ring Rdef(!) vanishabove dimension k.

Conjecture:

The non-trivial homotopy groups of Rdef(!) agree with H!(!),up to torsion.

The discrepancy in the torsion is meaningful: it has to do withthe question of which torsion classes in the cohomology can berealized as characteristic classes of (families of)representations.

Daniel A. Ramras Stable Representation Theory

Page 22: Stable Representation Theory - Mathematical Sciencesmath.iupui.edu/~dramras/UTEP.pdf · Cohomology and Classifying Spaces The cohomology groups Hk (G) can be defined Algebraically:

Concluding Remarks

For infinite discrete groups, the relationship betweenrepresentations and cohomology can be made

• explicit • geometric

Implications for other areas?

Each representation ! : G ! U(n) gives rise to a vector bundle

E! = (X " Cn)/G

!!X/G.

The geometry of bundles helps explain the relations betweenrepresentations and cohomology for infinite discrete groups,and may offer explanations for other related phenomena.

Daniel A. Ramras Stable Representation Theory