Lecture 18 - Stanford Universitysporadic.stanford.edu/quantum/lecture18.pdf · Lecture 18 Daniel...

Kashiwara Crystals Crystal bases and quantum groups Crystals of tableaux A peek at tableau combinatorics

Lecture 18

Daniel Bump

June 5, 2019

B(1) = 1 2 31 2 B(1,1) =1

2

1

3

2

32 1


Weight lattice

Let G be a complex semisimple or reductive Lie group with Liealgebra g. Let αi be the simple positive roots, and Λ the weightlattice. The index set is usually {1, · · · , r} so that αi are indexedby i ∈ I.

If G = GL(r) we will identify the weight lattice Λ = Zr withstandard basis ei. Then αi = ei − ei+1.


Categories of modules

Let G be a complex semisimple or reductive Lie group with Liealgebra g. When q is an indeterminate, the category offinite-dimensional integrable Uq(g)-modules is similar to thetheory of G modules. In both cases, representations arecompletely reducible, and there is a unique irreducible for everydominant weight; the dimensions of the irreducible modules ofUq(g) and G are equal. The main difference is that the categoryof Uq(g)-modules has an interesting braiding.

In previous lectures we considered what happens to modules ofUq(g) when we specialize q to a root of unity. Today we willconsider what happens when we specialize q→ 0.


Kashiwara crystals

The Hopf algebra Uq(g) does not make sense in the limit q→ 0,but if V is a Uq(g)-module, it is possible to slightly modify theaction of the Ei and Fi so that it is possible to make sense ofthem in the limiting case.

Then a very interesting combinatorial structure emerges. Let Λbe the weight lattice.

We define a crystal to be a set B with a map wt : B→ Λ. Thereare then assumed to exist maps ei : B→ B ∪ {0} andfi : B→ B ∪ {0} and εi,ϕi : B→ Z (for i in the index set) subjectto certain axioms.


Kashiwara crystals (continued)

It is assumed that if ei(x) = y is nonzero, then fi(y) = x. In thiscase

wt(ei(x)) = wt(x) + αi, wt(fi(y)) = wt(y) − αi.

Moreoverϕi(x) − εi(y) = 〈α∨

i ,wt(x)〉.

The crystal is called seminormal if

εi(x) = max{k|eki (x) 6= 0}, ϕi(x) = max{k|f k

i (x) 6= 0}.


Tableaux

Let G = GL(n) so that the weight lattice Λ = Zn.

If λ = (λ1, · · · , λn) is a partition, so λ1 > · · · > λn, asemistandard Young tableaux (SSYT) of shape λ is an array ofboxes filled by integers, in which the rows are weaklydecreasing, and the columns are strictly decreasing. If T is aSSYT we define the weight of T to be (µ1,µ2, · · · ,µn) where µi

is the number of entries in T equal to i. Thus

wt

1 1 2 3

2 3

3

= (2, 2, 3)


Representing crystals

If B is a crystal, we make B into a directed graph with edgeslabeled by the index set I. We draw an edge labeled by i from xto y if fi(x) = y. With this convention here are a couple of GL(3)crystals:

1 2 31 2 1

2

1

3

2

32 1

We require if fi(x) = y then wt(y) = wt(x) − αi. For example if

x =12, y =

13

then wt(x) = (1, 1, 0), wt(y) = (1, 0, 1), f2(x) = y,wt(x) − wt(y) = α2.


The character of a crystal

Let T be a maximal torus of G and z ∈ T. If B is a crystal, itscharacter is

χB(z) =∑x∈B

zwt(x).

Thus the characters of the crystals:

B(1) = 1 2 31 2 B(1,1) = 12

13

23

2 1

arez1 + z2 + z3, z1z2 + z1z3 + z2z3.

These are the characters of the irreducible representations ofGL(3) with highest weights (1, 0, 0) and (1, 1, 0).


Another example

B(2,1) =

2 33

2 23

1 33

1 23

1 13

1 32

1 22

1 12

2

1 2

1

2 1

1

2

χB(2,1) = x21x2 + x2

2x1 + x22x3 + x2

3x2 + x23x1 + x2

1x3 + 2x1x2x3


Tensor product

If B, C are crystals (with the same root system), there is acrystal B⊗ C. Our notation for the tensor product differs fromKashiwara’s but agrees with that of Bump and Schilling, CrystalBases: representations and combinatorics. As a set B⊗ C isthe Cartesian product of B and C, but we will denote theelement (x, y) as x⊗ y. We will denote wt(x⊗ y) = wt(x) + wt(y).

fi(x⊗ y) =

{fi(x)⊗ y if ϕi(y) 6 εi(x),x⊗ fi(y) if ϕi(y) > εi(x),

and

ei(x⊗ y) =

{ei(x)⊗ y if ϕi(y) < εi(x),x⊗ ei(y) if ϕi(y) > εi(x).

It is understood that x⊗ 0 = 0⊗ x = 0.


Tensor product (continued)

Alsoϕi(x⊗ y) = max

(ϕi(x),ϕi(y) + 〈wt(x),α∨

i 〉)

andεi(x⊗ y) = max

(εi(y), εi(x) − 〈wt(y),α∨

i 〉).

It may be checked that

B⊗ (C⊗D) ∼= (B⊗ C)⊗D.

The category of crystals is monoidal with the tensor operation.It also has a ⊕ operation, namely disjoint union and of course

A⊗ (B⊕ C) ∼= (A⊗ B)⊕ (A⊗ C).


Example: B(1) ⊗B(1)

3 ⊗ 3

3 ⊗ 2

3 ⊗ 1

2 ⊗ 3

2 ⊗ 2

2 ⊗ 1

1 ⊗ 3

1 ⊗ 2

1 ⊗ 1

1

2

2

2

12

1

1


Example (continued)

We see that the tensor product of the 3-dimensional crystalB(1) with itself decomposes into a 6 dimensional crystal withcharacter

x21 + x2

2 + x23 + x1x2 + x1x3 + x2x3

together with a copy of the crystal B(1,1).

This exactly mirrors the decomposition of the standardrepresentation π(1) of GL(3) tensored with itself:

π(1) ⊗ π(1) = π(2) ⊕ π(1,1)

where if λ is a partition, πλ denotes the irreduciblerepresentation of GL(3) with highest weight λ. π(2) and π(1,1)are the symmetric and exterior square representations.


Normal crystals

Let λ be a dominant weight. By Weyl character theory, G has aunique irreducible representation πλ with highest weight λ, withcharacter χλ.

Theorem (Kashiwara)There exists a family of crystals to be called normal with thefollowing properties.(1) There is a unique connected normal crystal Bλ for eachdominant weight λ.(2) A crystal is normal if and only if each connected componentis normal; and each connected component is a Bλ.(3) A tensor product of normal crystals is normal.(4) The character of Bλ equals χλ.


Implications

The theorem implies that the decomposition of crystals undertensor product exactly mirrors the decomposition ofrepresentations into irreducibles. Thus if

πλ ⊗ πµ =∑ν

cνλ,µπν

then cνλ,µ is also the multiplicity of Bν when Bλ ⊗Bµ isdecomposed into connected components.

This follows since the characters of irreducible representationsare linearly independent.


Methods of proof

Although Kashiwara arrived at crystals from the theory ofquantum groups, Littelmann arrived them independently fromconsiderations in algebraic geometry, related to theHodge-Seshadri-Lakshmibai theory of standard monomials.

Thus there were two independent proofs of the theorem, sinceLittelmann’s path-method proof was very different. It requiredproof that Littelmann’s crystals were the same as Kashiwara’s.

A purely combinatorial proof of the theorem was obtained byBump and Schilling. I believe it is also possible to prove it usingideas from geometric Langlands theory (MV polytopes).


Review: Uq(g)

For simplicity assume that the Cartan type is simply-laced. Letq be an indeterminate, so C(q) is the field of fractions of thepolynomial C[q]. Uq(g) had generators Ei, Fi and Ki

(i = 1, · · · , r) with Ki inqertible and relations:

KiEjK−1i = qaijEj, KiFjK−1

i = q−aijFj, [Ei,Fj] = δijKi − K−1

i

q − q−1 ,

and the quantum Serre relations: if αi and αj are orthogonalthen Ei and Ej commute,

E2i Ej − [2]qEiEjEi + EjE2

i = 0

where [2]q = q + q−1, and similar relations for the Fi. Let Uq(g)be the algebra generated by the Ei, Fi and Ki.


The comultiplication

We will deviate from Kashiwara’s notation for thecomultiplication, which is non-standard to begin with. This isrelated to the fact that our notation for the tensor product alsodiffers from Kashiwara’s. This does not change the crystalsthemselves.

We will take

∆(Ki) = Ki ⊗ Ki, ∆(Ei) = K−1i ⊗ Ei + Ei ⊗ 1,

∆(Fi) = 1⊗ Fi + Fi ⊗ K−1i .


Representations

There is a suitable category of representations Oint that iscompletely reducible, and generated by irreducible highestweight modules Vλ which look like the correspondingirreducibles L(λ) of G. That is, they have the same character.

Sometimes we wish to consider crystals of more generalrepresentations, such as Verma modules that live in the BGGCategory O.

Indeed Kashiwara’s theories make essential use of a crystalB(∞) that is the crystal of a Verma module. We will contentourselves with only explaining a part of what is actually neededand describe crystals for Oint.


Kashiwara operators

Kashiwara uses the notation ei and fi for the generators ofUq(g), which become operators on any module V. Then heuses ei and fi for certain modified Kashiwara operators.Instead, we use Ei and Fi for the generators and ei and fi for themodified Kashiwara operators.

We will writeV =

⊕µ

V(µ)

where V(µ) is the µ weight-space. We have

EiV(µ) ⊆ V(µ+ αi), FiV(µ) ⊂ V(µ− αi).


Kashiwara operators: construction

Using the divided factorials

F(n) =1

[n]q!Fn,

every element of V can be written uniquely as

x =

k∑m=0

F(m)(um)

where Ei(um) = 0, i.e. um is a multiple of the highest weightvector.To prove this, note that Ei,Fi,Ki generate a copy Uq(gi) ofUq(sl2). Hence this reduces to a statement about Uq(sl2)modules, which can be checked. Now define the Kashiwaraoperators

ei F(m)(u) = F(m−1)(u), fi F(m)(u) = F(m+1)(u)

for when Ei(u) = 0.


Crystal bases

Let A be the local ring consisting of f ∈ C(q) that have no poleat q = 0. Evaluating f at 0 gives a homomorphism A→ C soA/qA ∼= C.

By a lattice in a finite-dimensional C(q)-vector space V wemean an A-submodule L of V such that V ∼= C(q)⊗A L. Thismeans that L has an A-basis B that is also a C-basis of V.

Theorem (Kashiwara)V has a lattice L such that the Kashiwara operators ei and fimap L to itself. Hence ei and fi induce endomorphisms of thecomplex vector space L/qL. We may choose a basis B ofL/qL such that

ei(B) ⊂ B ∪ {0}, fi(B) ⊂ B ∪ {0}.


Crystal bases (continued)

The proof of Kashiwara’s theorem was very difficult.

The crystal base is a crystal as we defined crystals abstractly.Kashiwara also proved that if V and W are modules with crystalbases then V ⊗W has a crystal base, and he proved the tensorproduct rule that we gave earlier.

Crystal bases are the most natural context for discussingcertain topics in combinatorics, including work that was donebefore they were invented, such as theRobinson-Schensted-Knuth algorithm and deep work ofLascoux and Schützenberger.


Kashiwara-Nakashima

Kashiwara and Nakashima showed that the set of SSYT ofshape λ in {1, · · · , n} could be given the structure of a GL(n)crystal. All these crystals are subcrystals of tensor powers ofthe standard crystal.

1 2 · · · n1 2 n−1

We will describe these crystals of tableaux as subcrystals of⊗nB. Recall that wt(T) = (µ1, · · · ,µn where µi is the number ofi in the tableau T.


Crystals of rows

Let λ be a partition of r, having 6 n parts. Our goal is toorganize the tableaux of shape λ into a GL(n) crystal. First weconsider the case where λ = (r). Then a tableau of shape λ isjust a row of length r. The operation fi changes the rightmost ito i + 1 if i occurs in a tableau T, otherwise fi(T) = 0.

Example. If n = 4 and r = 2:

1 1

1 2 1 3

2 2 2 3 3 3

1

1

2

1

2 2


Crystals of rows as subcrystals of ⊗rB

If we identify the row a b · · · z of length r with

a ⊗ · · · ⊗ z ∈ ⊗rB

then we get the right tensor operation. It is important that

a 6 b 6 · · · 6 z.

Then the elements of ⊗rB of this form are closed under ei and fiand give a realization of the crystal of rows.


Crystals of columns

For crystals of columns, we recall that a column is strictlyincreasing. fi changes i to i + 1 but only if i + 1 does not alreadyoccur in the tableau T. If n = 4 and r = 2, B(1,1) looks like:

12

13

23

14

24

34

2

3

1

3

1 2


Crystals of columns as subcrystals of ⊗rB

If we identify a b · · · z with

z ⊗ · · · ⊗ a

then we get the right tensor operation. It is important that

a < b < · · · < z.

Then the elements of ⊗rB of this form are closed under ei and fiand give a realization of the crystal of columns.

The crystal B(r) of rows is the crystal of the r-th symmetricpower representation of GL(n).The crystal B(1r) = B(1,··· ,1) of columns is the crystal of ther-th exterior power representation of GL(n).


General Kashiwara-Nakashima crystals

If λ is an arbitrary partition of r of length 6 n and T is a SSYT ofshape λ then we identify T with an element of ⊗rB as follows.Read the entries of the tableau from left to right, from bottom totop.

1 1 2 2 2 32 2 34

=

4 ⊗ 2 ⊗ 2 ⊗ 3 ⊗ 1 ⊗ 1 ⊗ 2 ⊗ 2 ⊗ 2 ⊗ 3

It may be seen that ei and fi (as defined on ⊗rB) preserve theset of tableaux of shape λ embedded this way. This is the rowreading. There is another systematic way of embedding thecrystal of tableaux into ⊗rB that gives the same crystalstructure.


Review (from Lecture 7): Schur-Weyl-Jimbo duality

Schur-Weyl duality is a relationship between therepresentations of the symmetric group Sr and the generallinear group GL(n,C).

The groups Sr and GL(n,C) both act on the same vector space⊗rV where V = Cn, the standard module of GL(n). The groupGL(n) acts diagonally:

g(v1 ⊗ · · · ⊗ vn) = gv1 ⊗ · · · ⊗ gvn, g ∈ GL(n).

The symmetric group acts by permuting the components.

w(vi ⊗ · · · vr) = vw−1i ⊗ · · · vw−1r, w ∈ Sr.

The two actions obviously commute. The problem is todecompose ⊗rV into irreducibles.


Review: Irreducibles of GL(n) and Sr

Let λ be a partition of r of length 6 n. Thus λ = (λ1, λ2, · · · , λn)where λ1 > λ2 > · · · λn > 0 and

∑λi = r. Then λ indexes both

an irreducible representation of Sr and of GL(r,C).

The irreducible representations of Sr are indexed by partitionsof r. We will denote the corresponding irreducible of Sr as πSr

λ .

The irreducibles of GL(n,C) are indexed by dominant weights.A weight is a rational character of the diagonal subgroup; thegroup of weights is in bijection with Zn as follows: ifλ = (λ1, · · · , λn), then λ is called dominant if λ1 > · · · > λr. Inparticular, a partition of length 6 r is a dominant weight. Letπ

GL(n)λ be the corresponding irreducible.


Review: Schur-Weyl duality (concluded)

As a Sr ⊗ GL(n)-module, Schur-Weyl duality is the isomorphism

⊗rV ∼=⊕λ`r

πSrλ ⊗ π

GL(n)λ .

In Lecture 7 we studied Jimbo’s generalization, where GL(n) isreplaced by Uq(gln) and Sr is replaced by its Hecke algebra. Intoday’s lecture we will point out a combinatorial analog due toRobinson, Schensted, Knuth, Lascoux and Schützenberger.


The decomposition of ⊗rB

TheoremThe GL(r) crystal ⊗rB decomposes into crystals Bλ where λruns over the partitions of r of length 6 n. The number of timesBλ appears equals the degree of the irreducible representationπSrλ of Sr.

Needless to say there is much more to be said about thesematters, but I’ll stop here. These matters are central incombinatorics and predated crystals. But crystal theory is avery natural way of organizing tableaux combinatorics,including topics such as the plactic monoid, jeu de taquin, theLittlewood-Richardson rule and the famous RSK algorithm.

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