Representation theory through the lens of categorical … · Prolegomenon sl2 and geometry...

Prolegomenon sl2 and geometry Categorification from geometry Categorical sl2 actions

Representation theory through the lens of categoricalactions

Ben Webster

University of Virginia

June 3, 2015

Ben Webster UVA

Representation theory through the lens of categorical actions


My justification

So, the first question I should address is this: why should you show up?Why is what I’m about to tell you worth a few hours of your time, when youcould be at home petting your dog?

I’ll get into the specific content of in a moment, but let me put in a pitch forthe whole field of “categorification.”

Definition

cat·e·gor·i·fic·a·tion nounthe philosophy that any number worth its salt is the size of a set ordimension of a vector space, and any interesting set is the simple objects incategory (even if it doesn’t know it yet).

Ben Webster UVA



My justification

I think this idea tends to sound like a scam when people first hear it. Theytend to say things like “wait, but that isn’t going to be unique...”

I’m contractually obligated to reply “categorification is an art, not a functor.”

Sometimes, things in life aren’t unique, but they can still be useful.

Example

The rational function

x1001 x73

2 − x1002 x73

1 + x1002 x73

3 − x1003 x73

2 + x1003 x73

1 − x1001 x73

3

x21x2 − x2

2x1 + x22x3 − x2

3x2 + x23x1 − x2

1x3

is actually a polynomial with non-negative integral coefficients; Weyl tellsus that these are the weight multiplicities of a simple representation of GL3.

Might seem like a forced example, but who would have thought to look atSchur functions if they hadn’t known about GLn?

Ben Webster UVA



My justification

Another example is the replacement of Euler characteristic by homology.

homology is a stronger invariant of topological spaces than Eulercharacteristic, for sure.

much more important, it’s functorial! You can talk about maps betweensets or vector spaces, not between numbers.

While algebraic topology certainly does a good job of telling topologicalspaces apart, it would be pretty reductive to suggest that was its main goal. Italso gives us a lens for thinking about the structure of topology whichcouldn’t even be phrased just taking about numbers.

Ben Webster UVA



Themes

So these are the themes I want to emphasize throughout.

Categorifications:

allow us to show properties of weaker structures that might be hard tosee otherwise. Positivity and integrality results are especiallyimportant, but there are other examples.

suggest facets of already understood structures that are interesting ontheir own, but we might not have noticed. In particular, I maintain thatthe theory of quantum groups makes absolutely no sense unless youcategorify it.

allow us to ask questions about previously known invariants andstructures that simply made no sense before. Think Euler characteristicvs. homology.

Ben Webster UVA



Themes

In particular, I’m interested in categorified versions of structures in Lietheory, like universal enveloping algebras and particularly interestingrepresentations.

Lie theory has a lot of very rich structures (quantum groups, tensorproducts, canonical bases, braided categories, knot invariants, Fockspaces) so there are lots of examples to work with and structures to find.

many interesting categories have a Lie theoretic interpretation of theirdecategorification (e.g. reps of Fp[Sn] and slp à la LLT).

In particular, this gives us a new perspective on categories such asrepresentations of unitary groups over finite fields, category O for both Lie(super)algebras and Cherednik algebras.

Ben Webster UVA



sl2 and sets

Consider the Lie algebra g = sl2. This is the Lie algebra of 2× 2 trace 0matrices with the usual commutator.

If we let E =[

0 10 0

], F =

[0 01 0

]and H =

[1 00 −1

], then this algebra has a

presentation of the form

[H,E] = 2E [H,F] = −2F [E,F] = H

By definition, the universal enveloping algebra U(sl2) is the associativealgebra generated by the symbols E,F,H subject to the relations above(where [−,−] means commutator).

Ben Webster UVA



sl2 and sets

This algebra is fundamental for a lot of reasons, but let me emphasize aslightly less well-known one: it appears naturally from the structure of thecategory of finite sets.

Fix a finite set S, and consider the space V of functions 2S → C from thepower set of S. This is a C-vector space of dimension 2|S|.

Consider the operators E and F on V defined by

E · g(T) =∑

s∈S\T

g(T ∪ {s}) F · g(T) =∑t∈T

g(T \ {t})

A ridiculously easy, but still rather nice, theorem is:

Theorem

The operators E and F induce an action of sl2 on V. A function supportedon sets of size k has weight (H-eigenvalue) |S| − 2k.

Ben Webster UVA



sl2 and sets

This is one of these theorems that gets less and less impressive as you thinkabout it.

Proof. First, note that if S = S1 ∪ S2, then VS ∼= VS1 ⊗ VS2 , and that theendomorphisms E and F satisfy:

ES = E1 ⊗ 1 + 1⊗ E2 FS = F1 ⊗ 1 + 1⊗ F2.

Thus, if we have an sl2-action on VS1 and VS2 , then the action of VS is just thetensor product of these. Thus, we reduce to the case of a singleton: S = {s}.

In this case, V has a basis given by the characteristic functions v0 of ∅ and v1of {s}. In this case,

Ev0 = 0 Ev1 = v0 Fv1 = 0 Fv0 = v1.

This is how sl2 acts on C2, if we identify v0 7→ (1, 0) and v1 7→ (0, 1).

Ben Webster UVA



sl2 and sets

Ok, so this was just a fancy way of writing (C2)⊗|S|. But it’s not completelybankrupt; in fact, this is the kernel of the reason that sl2 has acategorification.

So, how can we upgrade this? Well, secretly, the power set 2S is secretly aunion of “Grassmannians over a field with 1 element.” Maybe that’s aninsane statement to make, but certainly any interesting game involvinginclusions of finite sets can be played using inclusions of vector spaces,especially of a finite field.

Let S be a vector space over a finite field Fp, let XS denote the union of theGrassmannians of subspaces in S (just as sets), and let VS denote the vectorspace of functions XS → C.

Ben Webster UVA



sl2 and sets

We can define some operators E and F on Vs much like before. Fornotational purposes, we set d = dim(S) and k = dim(W).

E · g(W) = p−k/2∑

W′⊃Wdim(W′/W)=1

g(W ′)

F · g(W) = pk−d/2∑

W′⊂Wdim(W/W′)=1

g(W ′)

Ok, some of you know what the next theorem is. Some of you think you do,but you’re wrong. These don’t define an action of sl2 in a literal sense,though perhaps they do in a moral one.

Ben Webster UVA



sl2 and sets

In this representation, [E,F] acts the functions concentrated in a singledimension (which thus are like weight spaces) by the scalar

[d − 2k]p1/2 =p1/2d−k − p−1/2d+k

p1/2 − p−1/2.

As a function of a formal variable q, we call [n]q = qn−q−n

q−q−1 the quantizationor quantum number of n; note that limq→1[n]q = n.

Theorem (Beilinson-Lusztig-Macpherson)

The operators E and F induce an action of the quantized universalenveloping algebra Uq(sl2), a flat deformation of U(sl2) over C[q, q−1],specialized at q = p1/2.

However, I refuse to define Uq(sl2) before defining its categorification, sodon’t worry too much about this difference.

Ben Webster UVA



sl2 and sets

The operations E and F can be thought of as having a geometric identitythemselves: for any finite set X, the set of functions C[X × X] has an algebrastructure under convolution:

f ? g(x1, x2) =∑x′∈X

f (x1, x′)g(x′, x2).

It’s easily shown that this is associative, and acts naturally on C[X] by

f ? h(x) =∑x′∈X

f (x, x′)h(x′).

Thus, the operations E and F are convolution with the functions

e(W,W ′) =

{p−1/2 dim W W ⊂ W ′, dim(W ′/W) = 10 otherwise

f (W,W ′) =

{p−1/2 dim(S/W) W ⊃ W ′, dim(W/W ′) = 10 otherwise

Ben Webster UVA



Cohomology of Grassmannians

I think you could rightly say “OK, that’s all lovely, but what does it have todo with the price of tea in China?” (By which I mean categorification..)

Again, I suspect some of you know what’s coming. As a hint, the set I’vecalled XS isn’t really just a set. It’s the Fp points of an algebraic variety. So,functions on it are avatars of sheaves, under the function-sheafcorrespondence.

So, in order to categorify, I should replace every function by a sheaf.

Or not.This is the first talk; maybe it’s a little early for the perverse sheaves.

Ben Webster UVA




I think you could rightly say “OK, that’s all lovely, but what does it have todo with the price of tea in China?” (By which I mean categorification..)

Again, I suspect some of you know what’s coming. As a hint, the set I’vecalled XS isn’t really just a set. It’s the Fp points of an algebraic variety. So,functions on it are avatars of sheaves, under the function-sheafcorrespondence.

So, in order to categorify, I should replace every function by a sheaf. Or not.This is the first talk; maybe it’s a little early for the perverse sheaves.

Ben Webster UVA




There is a decent replacement for them, though: the categories of modulesover cohomology rings. Let S be a complex vector space now, and letGr(k, S) be the Grassmannian. We’ll want to consider the categories ofmodules over the rings Hk := H∗(Gr(k, S);C) as a replacement for thevector space VS.

The maps E and F between these spaces should turn into functors,connected to the subspace

Gr(k ⊂ k + 1, S) = {(W,W ′) ∈ Gr(k, S)×Gr(k + 1, S)|W ⊂ W ′}.

Ben Webster UVA




We have a diagram of projections:

Gr(k ⊂ k + 1, S)

Gr(k, S) Gr(k + 1, S)

π1 π2

Definition

Let kHk+1 := H∗(Gr(k ⊂ k + 1, S)) as a left module over Hk via π∗1 and aright module over Hk+1 via π∗2 .

Let k+1Hk := H∗(Gr(k ⊂ k + 1, S)) with these actions reversed.

Consider the functors

Ek(M) := k−1Hk ⊗Hk M Fk(M) := k+1Hk ⊗Hk M.

Ben Webster UVA




Principle

The functors E and F categorify the operators E and F.

What the heck does this mean? I hope that intuitively, this seems reasonable:before, I had the characteristic function of a subvariety, and now I’vereplaced that with its cohomology, a categorically richer object.

Theorem (Beilinson-Lusztig-Macpherson?, Chuang-Rouquier)

kHk+1 ⊗Hk+1 k+1Hk ∼= (kHk−1 ⊗Hk−1 k−1Hk)⊕ H⊕d−2kk (k ≤ d/2)

(kHk+1 ⊗Hk+1 k+1Hk)⊕ H⊕2k−dk

∼= kHk−1 ⊗Hk−1 k−1Hk (k ≥ d/2)

That is, a module over Hk behaves like a vector of weight d − 2k under theaction of E and F.

Ben Webster UVA




Principle

The functors E and F categorify the operators E and F.

What the heck does this mean? I hope that intuitively, this seems reasonable:before, I had the characteristic function of a subvariety, and now I’vereplaced that with its cohomology, a categorically richer object.

Theorem (Beilinson-Lusztig-Macpherson?, Chuang-Rouquier)

Ek+1Fk ∼= Fk−1Ek ⊕ id⊕d−2k (k ≤ d/2)

Ek+1Fk ⊕ id⊕2k−d ∼= Fk−1Ek (k ≥ d/2)

That is, a module over Hk behaves like a vector of weight d − 2k under theaction of E and F.

Ben Webster UVA




Idea of proof

It’s a little difficult to give a proper proof without unpleasant formulae orgetting a bit more sheafy, but the underlying idea is simple.

First note that:

kHk+1 ⊗Hk+1 k+1Hk ∼= H∗(Gr(k ⊂ k + 1 ⊃ k, S)

kHk−1 ⊗Hk−1 k−1Hk ∼= H∗(Gr(k ⊃ k − 1 ⊂ k, S)

There’s a birational map between these varieties:

{W ′′ ⊃ W ′ ⊂ W} 7→ {W ′′ ⊂ W ′′ + W ⊃ W} when W ′′ 6= W

The “difference” between the two cohomologies is the same as the“difference” between the cohomologies of the bad sets for this map asmodules over H∗(Gr(k, S)).

Ben Webster UVA




One bad set is H∗(Gr(k ⊂ k + 1, S) and the other H∗(Gr(k ⊃ k − 1, S). Themaps

H∗(Gr(k ⊂ k+1, S)→ H∗(Gr(k, S)) H∗(Gr(k ⊂ k−1, S)→ H∗(Gr(k, S))

are fiber bundles with fiber Pd−k−1 in the first case and Pk−1 in the other.Thus, the cohomologies are free Hk-modules with ranks k and d − k.

Thus, the “difference” is

kHk+1 ⊗Hk+1 k+1Hk“− ”kHk−1 ⊗Hk−1 k−1Hk“ = ”(d − 2k) · Hk

Ben Webster UVA



Gradings

Thus far, everything I’ve done has been with ungraded modules, but thealgebras Hk are graded and we can make this whole constructionhomogeneous.

Definition

As a graded bimodules, let

k−1Hk := H∗(Gr(k ⊂ k + 1, S))(k − 1)

k+1Hk := H∗(Gr(k ⊂ k + 1, S))(d − k − 1)

where M(a) or qaM is M with its grading decreased by a.

Now, we should prove a graded version of the categorification theorem.

Ben Webster UVA



Gradings

Theorem (Beilinson-Lusztig-Macpherson?, Lauda)

Ek+1Fk ∼= Fk−1Ek⊕ id(d−2k)⊕id(d−2k−2)⊕· · ·⊕id(2k−d) (k ≤ d/2)

Ek+1Fk⊕ id(2k−d)⊕id(d−2k−2)⊕· · ·⊕id(d−2k) ∼= Fk−1Ek (k ≥ d/2)

That is, paying attention to the grading, we see that an Hk-module behaveslike a vector of weight d − 2k for the quantum group Uq(sl2). Setting q = 1corresponds to ignoring the grading.

This proof is almost the same as the ungraded proof; one just pays attentionto the grading on the cohomology of the bad loci. The term

[d − 2k]q = qd−2k + qd−2k−2 + · · ·+ q2k−d

thus comes from the difference of the cohomology of Pd−k−1 and Pk−1

(suitably shifted).Ben Webster UVA



Gradings

Theorem (Beilinson-Lusztig-Macpherson?, Lauda)

Ek+1Fk ∼= Fk−1Ek ⊕ id⊕[d−2k]q (k ≤ d/2)

Ek+1Fk ⊕ id⊕[2k−d]q ∼= Fk−1Ek (k ≥ d/2)

That is, paying attention to the grading, we see that an Hk-module behaveslike a vector of weight d − 2k for the quantum group Uq(sl2). Setting q = 1corresponds to ignoring the grading.

This proof is almost the same as the ungraded proof; one just pays attentionto the grading on the cohomology of the bad loci. The term

[d − 2k]q = qd−2k + qd−2k−2 + · · ·+ q2k−d

thus comes from the difference of the cohomology of Pd−k−1 and Pk−1

(suitably shifted).Ben Webster UVA



Upgrade to sheaves

Don’t worry too much if you’re not familiar with these notions, but it wouldbe a failure on my part not to mention how to think about this action in asomewhat more sophisticated way.

The maps we’ve introduced don’t just induce maps on cohomology; theyinduce functors between any category of sheaves satisfying Grothendieck’ssix-functor formalism. Most importantly on:

constructible derived categories on Gr(k, S) (in any formalism you like:analytic, étale Q`-sheaves, etc.) or

derived categories of D-modules on Gr(k, S).Of course, any one who knows the Riemann-Hilbert correspondence will tellyou these are essentially the same, but it’s nice to have both.

Ben Webster UVA



Upgrade to sheaves

In either case, we let

Ek(M) = (π1)∗π∗2M[k − 1] Fk(M) = (π2)∗π

∗1M[d − k − 1]

Gr(k ⊂ k + 1, S)

Gr(k, S) Gr(k + 1, S)

π1 π2

Theorem (Beilinson-Lusztig-Macpherson)

For constructible sheaves or D-modules, we have

Ek+1Fk ∼= Fk−1Ek⊕ id[d−2k]⊕ id[d−2k−2]⊕· · ·⊕ id[2k−d] (k ≤ d/2)

Ek+1Fk⊕ id[2k−d]⊕ id[d−2k−2]⊕· · ·⊕ id[d−2k] ∼= Fk−1Ek (k ≥ d/2).

Obviously, this is the “same” result we got for modules over thecohomology, but if there’s anything modern mathematics has taught me, it’sthat you have to be really careful about what “the same” means.

Ben Webster UVA



Definition

I really like this picture, but for the rest of you, it might feel a little limiting.I mean, how interesting are Grassmannians really?

This story does have a generalization to more general flag varieties, and evenquiver varieties, but those are still pretty boutique interests, and probably abit worrying for those of you who don’t like geometry.

Luckily, Chuang and Rouqiuer realized that you don’t really need all thisgeometry. You can boil it off and get a much more stripped down object.

The key ingredient we haven’t discussed is a nilHecke action.

Ben Webster UVA



Definition

For both of the frameworks we’ve described, we have

an endomorphism y : E→ E of the functor E given by the Chern classc1(W/W ′) on Gr(k ⊃ k − 1, S) = {(W,W ′) | W ′ ⊂ W}.an endomorphism ψ : E2 → E2 given by pushforward and pullback bythe map Gr(k ⊃ k − 1 ⊃ k − 2, S)→ Gr(k ⊃ k − 2, S).

A standard calculation on P1 shows that

ψ2 = 0 (y⊗ 1)ψ − ψ(1⊗ y) = ψ(y⊗ 1)− (1⊗ y)ψ = 1. (†1)

(ψ ⊗ 1)(1⊗ ψ)(ψ ⊗ 1) = (1⊗ ψ)(ψ ⊗ 1)(1⊗ ψ) (†2)We can represent these relations graphically by:

= + = 0 =

Ben Webster UVA



Definition

Thus, on Ek we have an action of:

Definition

The nilHecke algebra NHk is the algebra generated byy1, . . . , yk, ψ1, . . . , ψk−1, with relations:

[yi, yj] = 0 [ψm, ψn] = 0 |m− n| > 1

ψ2i = 0 [yi, ψj] = 0 i 6= j, j + 1

ψiψi+1ψi = ψi+1ψiψi+1 yiψi − ψiyi+1 = ψiyi − yi+1ψi = 1.

We can also think of this algebra geometrically representing the generatorsby

· · ·· · ·

yj

· · ·· · ·

ψj

and applying the local relations shown before.Ben Webster UVA



Definition

Still, now that we have two examples, we can make a definition.

Definition

An sl2-action on an additive category C consists of:

A direct sum decomposition C ∼= ⊕n∈ZCn.

Adjoint functors

En : Cn → Cn+2 and Fn : Cn → Cn−2

such that satisfy the relations

En−2Fn ∼= Fn+2En ⊕ id⊕n (n ≥ 0)

En−2Fn ⊕ id⊕−n ∼= Fn+1En (n ≤ 0)

natural transformations y : E→ E and ψ : E2 → E2 that satisfy therelations (†) (and thus define an action of NHk on Ek).

Ben Webster UVA



Definition

In order to talk in a clear way about categorification, we need the notion of aGrothendieck group. There are a few variations on this notion.

Definition

Given an additive category C, the split Grothendieck group (sGG) K(C) ofC is the abelian group generated by symbols [C] for C an object in C,modulo the relation:

[C1 ⊕ C2] = [C1] + [C2].

Let KC(C) ∼= K(C)⊗Z C.

Theorem

If C is an additive category with a weak sl2-action, the maps

[E] : KC(C)→ KC(C) [F] : KC(C)→ KC(C)

generate an sl2-action on KC(C).Ben Webster UVA



Definition

When confronted with a strange definition like this, one has to consider twothings:

1 are there actual examples of this definition? How easy is it to findinstances of it in nature?

2 what does it buy us to find out that some object we love is an instanceof this definition?

For #2, Chuang and Rouquier show that this is a powerful structure:

Theorem (Chuang-Rouquier)

For each highest weight, there is a unique universal abelian/additive

category which carries an sl2 action such that KC(C)sl2∼= Symn(C2). (This is

given by all/projective modules over Hk or its equivariant version.)

Every other abelian category with an sl2 action is filtered by copies of thisuniversal category. Every suitable property that holds for this basiccategorification also does for an arbitrary categorification.

Ben Webster UVA



Definition

In particular, we have a “categorified Weyl group action.”

Theorem (Chuang-Rouquier)

For any abelian category C with an sl2 action, we have a derivedequivalence Db(Cn) ∼= Db(C−n).

The nilHecke action is key here, since we need it to build the complex offunctors that gives the functor.

Ben Webster UVA



Symmetric groups

We’ll talk about more examples next time, but let me give one of the best.

Fix a field k, and consider the group algebras of the symmetric groups Sn

over k. These are related by induction and restriction functors. LetXm =

∑i<m(im) by the mth Jucys-Murphy element.

Proposition

The action of Xm on ResSmSm−1

M commutes with the Sm−1-action. That is, it is

a natural transformation of the functor ResSmSm−1

.

Fix a ∈ k, and let

Ea(M) = {h ∈ ResSmSm−1

M | (Xm − a)Nh = 0 for N � 0}.

Fa(M) = {s⊗ h ∈ IndSmSm−1

M | s(Xm − a)N ⊗ h = 0 for N � 0}.

Ben Webster UVA



Symmetric groups

Proposition (Lascoux-Leclerc-Thibon; Chuang-Rouquier)

For each a ∈ k, the functors Ea and Fa define a sl2-action; under thesefunctors, k as S0-module “generates” all Sn-modules.

The blocks of k[Sn] are precisely the modules that have the same weight foreach of these sl2’s; for each sl2, we get an involution of the set of blocks thatcorresponds to the reflection in the Weyl group of this sl2. The orbits of thegroup generated by all of these transformations are precisely the sets ofblocks that have the same defect group.

A corollary from the categorified Weyl group action is thus:

Theorem (Broué’s conjecture; Chuang-Rouquier)

If two blocks of k[Sn] have the same defect group, then they are derivedequivalent.

Ben Webster UVA



Next time

There are a number of directions we can go from here:there is a notion of an action of a higher rank Lie algebra (for example,k[Sn] -mod really carries an action of slp).there are a lot of interesting examples to dig into. Many situationswhere there are induction and restriction functors like for Sn fit into thisframework (and there are lots of those).a very rich vein here is the theory of canonical bases. Mostrepresentations with canonical bases (simples, tensor products, Fockspaces) have a categorification where the canonical bases correspond tonatural modules.by studying categorified tensor products, one can also categorifystructures like the braiding which are important for quantum topology.This leads to categorified knot invariants, like Khovanov homology andits generalizations.

I’ll touch on all of these themes next time, and in the next 3 lectures. I hopeto see you there.

Ben Webster UVA


Representation theory through the lens of categorical … · Prolegomenon sl2 and geometry...

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