higher representation theory in algebra and geometry ... et resumes cours...higher representation...

Higher representation theory in algebra and geometry:Lecture VI

Ben Webster

UVA

March 6, 2014

Ben Webster (UVA) HRT : Lecture VI March 6, 2014 1 / 26

References

For this lecture, useful references include:

Chuang and Rouquier, Derived equivalences for symmetric groups andsl2-categorification (introduces definition of categorical sln actions viaHecke algebras)

Khovanov and Lauda, A categorification of quantum sl(n) (coversconnection to partial flag varieties in great detail)

Brundan and Kleshchev, Blocks of cyclotomic Hecke algebras andKhovanov-Lauda algebras (introduced grading on symmetric groups andaffine Hecke algebras)

B. W., A note on isomorphisms between Hecke algebras (contains proofwe’ll discuss today)

The slides for the talk are on my webpage at:http://people.virginia.edu/~btw4e/lecture-6.pdf

You can also find some proofs that I didn’t feel like going through in class at:https://pages.shanti.virginia.edu/Higher_Rep_Theory/


http://people.virginia.edu/~btw4e/lecture-6.pdf

https://pages.shanti.virginia.edu/Higher_Rep_Theory/

Thus far...

We’ve talked a lot about categorical actions of sl2. We’ve seen them show upin the representation theory of symmetric groups, in the geometry ofGrassmannians and in the construction of knot invariants.

This leads us down a lot of interesting roads, but it’s not very satisfying. Afterall, sl2 is just the most basic of a big class of Lie algebras: the simple Liealgebras (or if you like to be fancier, the Kac-Moody algebras).

So, this is a natural direction to hunt. As with categorical sl2-actions, thething to look for is natural examples.


Simple Lie algebras

Recall, every simple Lie algebra over C (or more generally, Kac-Moodyalgebra) has a presentation by generators Ei,Fi,Hi for i ∈ I with relations

[Ei,Fi] = Hi [Hi,Ei] = 2Ei [Hi,Fi] = −2Fi

ad−aij+1Ei

Ej = 0 ad−aij+1Fi

Fj = 0 [Ei,Fj] = 0 [Hi,Hj] = 0

for some Cartan matrix A = (aij).

The most popular source of these is from graphs, where I is the vertex set, and−aij is the number of edges joining i and j.

The algebra sln+1 corresponds to a linear graph with n nodes.


Examples

Theorem (Beilinson-Lusztig-Macpherson)

There is a natural U√p(sln) action on the space of functions on the set of alln− 1-step flags in a fixed vector space S, such that:

the maps Ei and Fi are induced by the pullback and pushforward areinduced by the correspondence

Gr(· · · ⊂ ki ⊂ · · · , S)

Gr(· · · ⊂ ki − 1 ⊂ ki ⊂ · · · , S)

Gr(· · · ⊂ ki − 1 ⊂ · · · , S) π1

π2

the functions concentrated on Gr(kn−1 ⊂ kn−2 ⊂ · · · ⊂ k1, S) are ofweight (d − k1, k1 − k2, . . . , kn−1).

the functions constant on each Gr(kn−1 ⊂ kn−2 ⊂ · · · ⊂ k1, S) form acopy of the simple representation Symd(Cn).


Symmetric groups

Fix a field k, and let K = ⊕mK(k[Sm] -proj) be the Grothendieck group ofprojective k[Sm] modules.

Earlier, we defined functors Fa and Ea for a ∈ Z by

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}.

The functors Ei and Fi induce operators Ei and Fi on K.

Proposition

Each (Ei,Fi) generate a copy of sl2, and

[Ei, [Ei,Ei±1]] = 0 [Ei,Ej] = 0 (i 6= j± 1)

[Ei,Fj] = 0 (i 6= j) [Fi, [Fi,Fi±1]] = 0 [Fi,Fj] = 0 (i 6= j± 1)


Symmetric groups

That is, we have an action of the Lie algebra whose Dynkin diagram hasvertices given by the image of Z in k where i and i + 1 are adjacent.

When char(k) = 0, then this is sl∞.

When char(k) = p, then this is the affine Lie algebra slp.

Proposition

The Grothendieck group K is an irreducible representation of sl∞ or slp,generated by a highest weight vector satisfying Hiv = δi,0v (“the basicrepresentation”).


Categorifications of slnThese examples suggest there’s some underlying structure here, so let’s findit. The place that’s most conducive to doing the calculation is on flagvarieties. Recall that:

Proposition

The cohomology Gr(kn−1 ⊂ kn−2 ⊂ · · · ⊂ k1) is isomorphic to Ckn−1,...,k1d ,

generated by ei(1, kn−1), ei(kn−1 + 1, kn−2), . . . , ei(k1 + 1, d) in thecoinvariant algebra Cd.

Whatever a categorical action of sln is, we have an example of one on thecategories ⊕kCkn−1,...,k1

d -mod with functors Ei and Fi given by the tensorproducts

Ei = C...,ki−1,ki,...d ⊗

C...,ki,...d

− : C...,ki,...d -mod→ C...,ki−1,...

d

Fi = C...,ki,ki+1,...d ⊗

C...,ki,...d

− : C...,ki,...d -mod→ C...,ki+1,...

d


Categorifications of sln

First of all, the functors Ei and Fi for each i generate a categorical action ofsl2. That is Ei and Fi satisfy sl2 relations up to isomorphism, are biadjoint,and Fm

i has an action of the nilHecke algebra.

Well, that’s a good start, but presumably it’s not enough; we need to havesome kind of interaction between these different sl2’s.

Thinking geometrically, FiFj corresponds to tensor product with H∗ of thespace of triples of flags

Xi,j = {(V•,V ′•,V′′• ) | Vk ⊃ V ′k, dim(Vk/V ′k) = δk,i,

V ′m ⊃ V ′′m, dim(V ′m/V ′′m) = δm,j}

If i 6= j± 1, then switching the order of i and j gives the same space (Exercise:write down the isomorphism), and so FiFj ∼= FjFi.


Categorifications of sln

However, if j = i + 1, it’s a different story. We may as well forget all the otherspaces in the flag, and assume that n = 3, i = 1 and j = 2. Thus, we have that

X1,2 = {V1 ⊃ V2,V ′′1 ⊃ V ′′2 }

X2,1 = {V1 ⊃ V ′′1 ⊃ V2 ⊃ V ′′2 }

Thus, we have an obvious inclusion map X2,1 → X1,2 and thus pullback andpushforward maps in cohomology.

These maps aren’t isomorphisms. Instead, their composition in eitherdirection is multiplication by the Euler class of the line bundleHom(V2/V ′′2 ,V1/V ′′1 ) since X2,1 is the vanishing set of a section of this bundle(the induced canonical map).


KLR algebras

In the case of sl2, the structure of a categorical action is controlled by thenilHecke action on powers of the functor F. What will this structure look likein the sln case?

Consider the sum F := ⊕Fi. What acts on Fk?

there are idempotents ei which project to Fik · · ·Fi1 .

the action of dots y1, . . . , yk induced by the individual sl2 actions.elements ψj which acts on Fik · · ·Fi1 by

1 the Demazure operator if ij = ij+12 the pushforward by the map Xij+1,ij → Xij,ij+1 if ij+1 = ij + 13 the pullback by the map Xij,ij−1 → Xij−1,ij if ij+1 = ij − 14 the map induced the isomorphism Xij+1,ij

∼= Xij+1,ij if ij+1 6= ij, ij ± 1.


Diagrams

We represent these with diagrams much like before.

i1 i2 in

· · ·

ei

deg = 0

i1 ij in

· · ·· · ·

yj

deg = 2

i1 ij ij+1 in

· · ·· · ·

ψj

deg = −〈αij , αij+1〉

DefinitionThe Khovanov-Lauda-Rouquier (KLR) or quiver Hecke algebra Rm for slnis the algebra generated by these elements with m strands modulo therelations on the next slide.


Diagrams

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i− 1

=

i− 1i i− 1i

-

i i + 1

=

i + 1i i + 1i

-

ki j

=

ki j

unless i = k = j± 1

i i

= 0

ii i− 1

=

ii i− 1

+

ii i− 1

ii i + 1

=

ii i + 1

−

ii i + 1


Categorical actions of sln

DefinitionA categorical action of sln is a collection of categorical sl2 actions (Fi,Ei) fori = 1, . . . , n− 1, together with an extension of the nilHecke action to anaction of Rm on (⊕Fi)

m.

We’ve deliberately set things up so we know at least one example of these: themodules over Ckn−1,...,k1

d for all different choices of k.


Serre relations

Why do the Serre relations hold? Consider the relation

ii i− 1

=

ii i− 1

−

ii i− 1

This is splitting of a sum as two orthogonal idempotents. In fact, this showsdirectly that:

FiFi−1Fi ∼= F(2)i Fi−1 ⊕ Fi−1F

(2)i

which is just a rearranged Serre relation!

PropositionAny categorical sln action actually does induce an sln action on theGrothendieck group. (Note, I haven’t checked that [Ei,Fj] = 0.)


Categorical actions for other Lie algebras

It’s relatively easy to guess how to extend this to the case of sln.

You can construct a KLR algebra for sln with the same sort of diagramsand relations, but with strands labeled by Z/nZ, rather that{1, . . . , n− 1}.A categorical sln-action is a collection of sl2 actions which have anaction of this KLR algebra on (⊕Ei)

m.

Now, two things should bother you about this:1 Many of you probably know that an sln or sln was defined in a different

way in Chuang-Rouquier, using an affine Hecke algebra instead.2 It’s not clear that there are any examples of categorical sln-actions “in the

wild” though you may recall that symmetric groups looked promising.


Hecke vs. KLR

There’s an alternate definition of categorical actions via affine Hecke algebras.

DefinitionThe degenerate affine Hecke algebra hm of rank m is generated byt1, . . . , tm−1, x1, . . . , xm with relations:

t2i = 1 titi±1ti = ti±1titi±1 titj = tjti (i 6= j± 1)

xixj = xjxi tixiti = xi+1 − ti xitj = tixj (i 6= j, j + 1)

DefinitionThe affine Hecke algebra Hm(v) of rank m is generated by T1, . . . ,Tm−1,X±1

1 , . . . ,X±1m with relations:

(Ti + 1)(Ti − v) = 0 TiTi±1Ti = Ti±1TiTi±1 TiTj = TjTi (i 6= j± 1)

XiXj = XjXi TiXiTi = vXi+1 XiTj = TiXj (i 6= j, j + 1)


Hecke vs. KLR

Consider a pair of adjoint functors (F,E) such that Fm carries an action of theaffine Hecke algebra over k for some v ∈ k. Let U ⊂ k be the union of thespectra of X acting on F (and that F(M) is the sum of its generalizedeigenspaces). We give U a graph structure by adding an edge u1 → u2 ifu2 = vu1.

We can decompose F = ⊕u∈UFu into its generalized eigenspaces for X, andsimilarly E = ⊕u∈UEu.

Alternate definitionIf the functors Ei and Fi define a weak sl2 action, then the functors Ei and Fi

define an action of gU , the algebra with Dynkin diagram U.

Since U is a union of linear and cyclic graphs, this just means a bunch ofcommuting slr and sls actions.


Hecke vs. KLR

Consider a pair of adjoint functors (F,E) such that Fm carries an action of thedegenerate affine Hecke algebra over k. Let u ⊂ k be the union of the spectraof x acting on F (and that F(M) is the sum of its generalized eigenspaces).We give u a graph structure by adding an edge u1 → u2 if u2 = u1 + 1.

We can decompose F = ⊕u∈UFu into its generalized eigenspaces for x, andsimilarly E = ⊕u∈UEu.

Alternate definitionIf the functors Ei and Fi define a weak sl2 action, then the functors Ei and Fi

define an action of gU , the algebra with Dynkin diagram U.

Since U is a union of linear and cyclic graphs, this just means a bunch ofcommuting slr and sls actions.


Hecke vs. KLR

For example, we have an action of hm on IndSk+mSk

with

ti 7→ (i + k, i + k + 1) xi 7→∑g<i

(g, i)

The spectrum u is the image of Z in k; thus, as we saw before, it’s a cycle ifchar(k) = p, and an infinite line if char(k) = 0.

Proposition

This defines a categorical slp or sl∞-action on ⊕mk[Sm] -mod (according toour alternate definition).

Similarly, we’ll get an slp-action if we look at representations of the (finite)Hecke algebra with v ∈ k a primitive pth root of unity. This is just one aspectof the analogy of this case with the case char(k) = p.


Hecke vs. KLR

So, why are these “the same thing”? Well, of course, it would be great if the(d)AHA were isomorphic to a KLR algebra, but a moment’s thought showsthis can’t be so (for which graph?).

Proposition

Instead, there’s a natural completion of Hm(v) at the collection of quotientswhere Xi has spectrum U which is isomorphic to the KLR algebra Rm withgraph U completed according to its grading.

For example, if we let U be the pth roots of unity in k of characteristic primeto p, we’ll get the KLR algebra for slp.


Hecke vs. KLR

So, why are these “the same thing”? Well, of course, it would be great if the(d)AHA were isomorphic to a KLR algebra, but a moment’s thought showsthis can’t be so (for which graph?).

PropositionInstead, there’s a natural completion of hm at the collection of quotientswhere xi has spectrum u which is isomorphic to the KLR algebra Rm withgraph u completed according to its grading.

For example, if we let u be Z/pZ in k of characteristic p, we’ll get the KLRalgebra for slp.


Polynomial representations

In order to think about KLR algebras, one of the most useful tools is arepresentation on a sum of polynomial rings. Let I be the vertex set of anoriented graph. Consider the sum of polynomial rings ⊕i∈Imk[Y1, . . . ,Ym]εi.Consider the operators

ei · f (Y1, . . . ,Ym)εj = f (Y1, . . . ,Ym)δi,jεj

yk · f (Y1, . . . ,Ym)εj = Ykf (Y1, . . . ,Ym)εj

ψk · f (Y1, . . . ,Ym)εj =

{f−f si

Yk+1−Ykεsk·j jk = jk+1

(Yk+1 − Yk)#{jk→jk+1}f skεsk·j jk 6= jk+1

PropositionThe algebra these generate is the KLR algebra of I.

Generally, we can choose a Cartan matrix A and polynomials Pij(u, v) with

Pij(u, v)Pji(v, u) = tiju−aji + · · ·+ tjiv−aij .



In order to think about KLR algebras, one of the most useful tools is arepresentation on a sum of polynomial rings. Let I be the vertex set of anoriented graph. Consider the sum of polynomial rings ⊕i∈Imk[Y1, . . . ,Ym]εi.Consider the operators

ei · f (Y1, . . . ,Ym)εj = f (Y1, . . . ,Ym)δi,jεj

yk · f (Y1, . . . ,Ym)εj = Ykf (Y1, . . . ,Ym)εj

ψk · f (Y1, . . . ,Ym)εj =

{f−f si

Yk+1−Ykεsk·j jk = jk+1

Pikik+1(Yk+1,Yk)f skεsk·j jk 6= jk+1

PropositionThe algebra these generate is the KLR algebra of I.

Generally, we can choose a Cartan matrix A and polynomials Pij(u, v) with

Pij(u, v)Pji(v, u) = tiju−aji + · · ·+ tjiv−aij .



The affine Hecke algebra also has a (signed) polynomial representation onk[X±1

1 , . . . ,X±1m ] given by

TiF(X1, . . . ,Xm) = −Fsi + (1− q)Xi+1Fsi − F

Xi+1 − Xi

We’ll also complete this representation at the quotients where the spectrum ofXi has spectrum U. For each u = (u1, . . . , um), we have the εu of 1 to thegeneralized ui-eigenspace for Xi.

The idea of the isomorphism is to match up these two representations aftercompletion. Fix any power series B(z) = 1 +

∑∞i=1 Bizi with B1 6= 0.

Xa11 · · ·X

amm εu ↔ (u1B(y1))a1 · · · (umB(ym))amεu



The degenerate affine Hecke algebra also has a (signed) polynomialrepresentation on k[x1, . . . , xm] given by

tif (x1, . . . , xm) = −f si − f si − fxi+1 − xi

We’ll also complete this representation at the quotients where the spectrum ofxi has spectrum u. For each u = (u1, . . . , um), we have the εu of 1 to thegeneralized ui-eigenspace for xi.

The idea of the isomorphism is to match up these two representations aftercompletion. Fix any power series b(z) = b1z + . . . with b1 6= 0.

xa11 · · · x

amm εu ↔ (b(y1) + u1)a1 · · · (b(ym) + um)amεu



The trick is to consider the “intertwining elements”

Φr := Tr +∑

u s.t. ur 6=ur+1

1− qeh

1− XrX−1r+1

eu +∑

u s.t. ur=ur+1

eu

If you look at how this element acts in the polynomial representation, you canmatch it up with the element ψr:

Φr · F =

urB(yr)− vur+1B(yr+1)

ur+1B(yr+1)− urB(yr)Fsrεusr ur 6= ur+1

B(yr)− vB(yr+1)

B(yr+1)− B(yr)(Fsr − F)εu ur = ur+1

We then use the fact that B(yr)− aB(yr+1) = (1− a) + B1(yr − yr+1) + · · ·is invertible if a 6= 1, whereas (B(yr)− B(yr+1))/(yr − yr+1) = B1 + · · · iswell-defined and invertible.



The trick is to consider the “intertwining elements”

φr := tr +∑

u s.t. ur 6=ur+1

1xi − xi+1

eu +∑

u s.t. ur=ur+1

eu

If you look at how this element acts in the polynomial representation, you canmatch it up with the element ψr:

Φr · F =

b(yr) + ur − b(yr+1)− ur+1 − 1

b(yr+1) + ur+1 − b(yr)− urFsrεusr ur 6= ur+1

b(yr)− b(yr+1)− 1b(yr+1)− b(yr)

(Fsr − F)εu ur = ur+1

We then use the fact that b(yr)− b(yr+1) + a = a + b1(yr − yr + 1) + · · · isinvertible if a 6= 0, whereas (b(yr)− b(yr+1))/(yr − yr+1) = b1 + · · · iswell-defined and invertible.



That is, we send

Φreu 7→

(B(yr)− vB(yr+1))(yr − yr+1)

B(yr+1)− B(yr)ψreu ur = ur+1

B(yr)− B(yr+1)

(v−1B(yr+1)− B(yr))(yr+1 − yr)ψreu ur = vur+1

urB(yr)− vur+1B(yr+1)

ur+1B(yr+1)− urB(yr)ψreu ur 6= ur+1, vur+1

TheoremThis induces an isomorphism of completions between the KLR algebra R forU and Hm(v).

Note that this isomorphism is highly non-unique. I kind of like to useB(x) = ex, but Brundan and Kleshchev use B(x) = 1 + x.



That is, we send

φreu 7→

(b(yr)− b(yr+1)− 1)(yr − yr+1)

b(yr+1)− b(yr)ψreu ur = ur+1

b(yr)− b(yr+1)

(b(yr+1)− b(yr)− 1)(yr+1 − yr)ψreu ur = ur+1 + 1

b(yr) + ur − b(yr+1)− ur+1 − 1b(yr+1) + ur+1 − b(yr)− ur

ψreu ur 6= ur+1, ur+1 + 1

TheoremThis induces an isomorphism of completions between the KLR algebra R for uand hm.

Note that this isomorphism is highly non-unique. I kind of like to useB(x) = ex, but Brundan and Kleshchev use B(x) = 1 + x.


Gradings on k[Sm]

One really interesting consequence of this isomorphism is that k[Sm] has anunexpected grading:

Proposition

The group algebra k[Sm] is the quotient of hm by x1 = 0. The other xi’s go tothe Jucys-Murphy elements.

Thus, we can also describe k[Sm] as a quotient of the KLR algebra by the twosided ideal generated by ei with i1 6= 0, and y1.

For example, if m = 2, then as long as p > 2, this quotient is spanned by e0,1and e0,p−1, the idempotents projecting to the trivial and sign reps. If p = 2,then these are the same, and k[S2] = k[y2]/(y2

2).


Gradings on k[Sm]

If interpreted correctly, most natural modules over k[Sm] such as simples,Specht modules, permutation modules, and projectives can be graded.

Thus, interesting questions like:(∗) what are the multiplicities of simples modules in Spechts/projectives?have q-analogues where we consider graded multiplicities. These have, forexample, interesting connections to Kazhdan-Lusztig polynomials.

To be precise, if one looks instead at the (finite) Hecke algebra over C atv = e2πi/p, the graded multiplicities of simples in Specht modules will beaffine parabolic Kazhdan-Lusztig polynomials. The multiplicities over Fp arenot the same (though they are bounded below by the char 0 multiplicities) forall p, but for p in some range c(m) ≤ p ≤ m.

A long-standing conjecture of James had been that c(m) ≤ √p. This wasrecently disproven by Williamson.


higher representation theory in algebra and geometry ... et resumes cours...higher representation...

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