Higher representation theory in algebra and geometry: Lecture II

Higher representation theory in algebra and geometry: Lecture II Ben Webster UVA

February 10, 2014

Ben Webster (UVA)

HRT : Lecture II

February 10, 2014

1 / 35

References

For this lecture, useful references include: Chuang and Rouquier, Derived equivalences for symmetric groups and sl2 -categorification (introduces notion of strong sl2 categorification and covers several examples, such as the symmetric group) Lauda, A categorification of quantum sl(2) and Categorified quantum sl(2) and equivariant cohomology of iterated flag varieties (discusses the relationship with the cohomology of Grassmannians and is the source of the pictures) The slides for the talk are on my webpage at: http://people.virginia.edu/ btw4e/lecture-2.pdf

Ben Webster (UVA)

HRT : Lecture II

February 10, 2014

2 / 35

Reminder about last time So, last time we discussed the Lia algebra sl2 , its quantized universal enveloping algebra Uq (sl2 ) and how we might categorify them. The underlying idea was to look at these objects in a way that makes the categorification of them very natural, in connection with Grassmannians. In particular, the generators E and F were connected to the diagram π1 Gr(k ⊂ k + 1, S) π2 Gr(k, S)

Gr(k + 1, S)

We first interpreted this in terms of counting points, but then decided we could “upgrade” from this to considering cohomology or push and pull for sheaves on this diagram. Ben Webster (UVA)

HRT : Lecture II

February 10, 2014

3 / 35

Cohomology and bimodules Hk := H ∗ (Gr(k, S)) k−1 Hk

:= H ∗ (Gr(k − 1 ⊂ k, S))(k − 1)

Ek (M) := k−1 Hk ⊗Hk M

k+1 Hk

:= H ∗ (Gr(k + 1 ⊃ k, S))(d − k − 1)

Fk (M) := k+1 Hk ⊗Hk M.

Theorem Ek+1 Fk ∼ = Fk−1 Ek ⊕ id⊕[d−2k]q

(k ≤ d/2)

Ek+1 Fk ⊕ id⊕[2k−d]q ∼ = Fk−1 Ek

(k ≥ d/2)

So, this is the relations of (quantum) sl2 in a categorified form. But as a definition, we decided this was unsatisfying; we wouldn’t know if we found other functors satsifying the same isomorphisms if we were really seeing the same thing. Ben Webster (UVA)

HRT : Lecture II

February 10, 2014

4 / 35

Natural transformations

So, the thing that’s missing is some understanding of what these isomorphisms are, but more generally of maps between functors. There’s an obvious set of questions we haven’t looked at at all: If I have two monomials in the functors F and E, what are the natural transformations between them? If you don’t like natural transformations, you can just think of these as maps between the tensor products of bimodules.

Ben Webster (UVA)

HRT : Lecture II

February 10, 2014

5 / 35

Cohomology of Grassmannians and partial flag varieties If we want to get our hands dirty, we’ll have to actually think a bit about the cohomology of Grassmannians. Of course, the Grassmannian carries a tautological vector bundle T ⊂ Gr(k, S) × S whose fiber over a point is the vector space itself; there’s also the quotient S/T of the trivial bundle with fiber S by T. This gives us a bunch of cohomology classes, by taking the Chern classes of these bundles xi = ci (T) wi = ci (S/T). By the Whitney sum formula, these satisfy the relation (1+c1 (T)t+c2 (T)t2 +· · · )(1+c1 (S/T)t+c2 (S/T)t2 +· · · ) = 1+c1 (S)t+· · · .

Ben Webster (UVA)

HRT : Lecture II

February 10, 2014

6 / 35

Cohomology of Grassmannians and partial flag varieties If we want to get our hands dirty, we’ll have to actually think a bit about the cohomology of Grassmannians. Of course, the Grassmannian carries a tautological vector bundle T ⊂ Gr(k, S) × S whose fiber over a point is the vector space itself; there’s also the quotient S/T of the trivial bundle with fiber S by T. This gives us a bunch of cohomology classes, by taking the Chern classes of these bundles xi = ci (T) wi = ci (S/T). By the Whitney sum formula, these satisfy the relation (1 + x1 t + x2 t2 + · · · )(1 + w1 t + w2 t2 + · · · ) = 1.

Ben Webster (UVA)

HRT : Lecture II

February 10, 2014

6 / 35

Cohomology of Grassmannians and partial flag varieties So, for example: 1=1 You can solve for xi ’s in terms of wi ’s or vice versa.

Ben Webster (UVA)

HRT : Lecture II

February 10, 2014

7 / 35

Cohomology of Grassmannians and partial flag varieties So, for example: x1 + w1 = 0 You can solve for xi ’s in terms of wi ’s or vice versa.

Ben Webster (UVA)

HRT : Lecture II

February 10, 2014

7 / 35

Cohomology of Grassmannians and partial flag varieties So, for example: x2 + x1 w1 + w2 = 0 You can solve for xi ’s in terms of wi ’s or vice versa.

Ben Webster (UVA)

HRT : Lecture II

February 10, 2014

7 / 35

Cohomology of Grassmannians and partial flag varieties So, for example: x3 + x2 w1 + x1 w2 + w3 = 0 You can solve for xi ’s in terms of wi ’s or vice versa.

Ben Webster (UVA)

HRT : Lecture II

February 10, 2014

7 / 35

Cohomology of Grassmannians and partial flag varieties So, for example: x3 + x2 w1 + x1 w2 + w3 = 0 You can solve for xi ’s in terms of wi ’s or vice versa. Theorem The cohomology H ∗ (Gr(k, S)) of the Grassmannian is the quotient of the polynomial ring C[x1 , . . . , xk , w1 , . . . , wd−k ] by these Whitney sum relations. Alternatively, you can consider an infinite polynomial ring C[x1 , . . . , w1 , . . . ] modulo the Whitney sum relation, and write all the cohomologies of all the Grassmannians as quotients of this by setting xi = 0

Ben Webster (UVA)

(i > k)

wj = 0

HRT : Lecture II

(j > d − k).

February 10, 2014

7 / 35

Natural transformations This also handled the question of one set of natural transformations: Proposition The natural transformations of the identity functor idk are the ring H ∗ (Gr(k, S)) itself. OK, that was easy. What about Ek or Fk ? The ring H ∗ (Gr(k ⊂ k + 1, S)) can be found in a very similar way. But now, we have k and k + 1 dimensional tautological bundles T k , T k+1 , and a line bundle T k+1 /T k . The notation can get confusing, since I have a class I was calling xi on H ∗ (Gr(k, S)) and on H ∗ (Gr(k + 1, S)), and these don’t pullback to the same place: one goes to ci (T k ) and the other to ci (T k+1 ). Ben Webster (UVA)

HRT : Lecture II

February 10, 2014

8 / 35

Cohomology and symmetric polynomials In the interest of not going crazy, let’s look at this in a different way: let z1 , . . . , zk be the Chern roots of T k , and zk+1 , . . . , zd be the Chern roots of S/T k . For us these are just formal symbols, which satisfy ei (1, k) = ei (z1 , . . . , zk ) = ci (T k )

ei (k + 1, d) = ei (zk+1 , . . . , zd ) = ci (S/T k )

In the formulation with Chern roots, the Whitney sum relations simply say that the Chern roots of S are the union of those for T k and S/T k . The rest follows from the fact that: ei (1, d) = ei (1, k) + ei−1 (1, k)e1 (k + 1, d) + · · · + ei (k + 1, d). The Whitney sum relations for the Grassmannian simply say that ei (1, d) = 0 for all i > 0. Ben Webster (UVA)

HRT : Lecture II

February 10, 2014

9 / 35

Cohomology and symmetric polynomials Definition The coinvariant algebra Cd is the quotient of C[z1 , . . . , zd ] by the relations ei (1, d) = 0. This inherits an Sd action permuting the variables. k ,...,kp

Let Sm1 ,...,mp := Sm1 × · · · × Smp ⊂ Sd , and Cd1

Sk

= Cd 1

,k2 −k1 ,k3 −k2 ,...,d−kp

.

Proposition The algebra H ∗ (Gr(k, S)) is isomorphic to Cdk which is generated by the elementary symmetric functions ei (1, k) and ei (k + 1, d). This can be generalized to the partial flag variety Gr(k1 ⊂ k2 ⊂ · · · ⊂ kp ). Proposition k ,k2 ,...,kp

The cohomology Gr(k1 ⊂ k2 ⊂ · · · ⊂ kp ) is isomorphic to Cd1 generated by ei (1, k1 ), ei (k1 + 1, k2 ), . . . , ei (kp + 1, d) in Cd . Ben Webster (UVA)

HRT : Lecture II

,

February 10, 2014

10 / 35

Cohomology and symmetric polynomials Thus, the bimodule k+1 Hk can be thought of as Cdk,k+1 , with the left action by Hk+1 = Cdk+1 and the right action by Hk = Cdk . Proposition Since Hk+1 and Hk generate Cdk,k+1 , the bimodule maps from k+1 Hk to itself are given by multiplication by elements of Cdk,k+1 . Visually, we can represent these elements using diagrams: wj

xi F

The left/right sides correspond to the left/right action, the bottom to the source bimodule and the top to the target. I represent the action of zk+1 by a dot on the line. Ben Webster (UVA)

HRT : Lecture II

February 10, 2014

11 / 35

Diagrams These diagrams have the huge advantage that I don’t confuse the left and right actions. I can write a relation like ci (T k+1 ) = ci (T k ) + c1 (T k+1 /T k )ci (T k ): =

xi F

xi F

+

xi−1 F

without making a mess separating the left and right actions. In general, I can describe any element of a tensor product of k±1 Hk ’s using a diagram like this, with a line for each term in the tensor product. wj

xi F Ben Webster (UVA)

E HRT : Lecture II

F February 10, 2014

12 / 35

More natural transformations This describes all elements, but not all bimodule maps. For example, if d = 2, then 2 H1 ⊗ 1 H0 ∼ = C[t]/(t2 ) ⊗C[t]/(t2 ) C[t]/(t2 ) ∼ = C2 , so we must get the matrices acting on here somehow. In general, k+2 Hk+1 ⊗ k+1 Hk is always two copies of the same module.

Gr(k, S)

Gr(k ⊂ k + 1 ⊂ k + 2, S) π1,3 π1 π2 Gr(k ⊂ k + 2, S)

Gr(k + 2, S)

The map π1,3 is a fiber bundle with fiber P1 . Thus as a bimodule: k+2 Hk+1 ⊗ k+1 Hk

Ben Webster (UVA)

∼ = H ∗ (Gr(k ⊂ k + 1 ⊂ k + 2, S)) ∼ = H ∗ (Gr(k ⊂ k + 2, S))⊕2 HRT : Lecture II

February 10, 2014

13 / 35

Demazure operators From the perspective of polynomials, this is also easy to see: if I consider the invariants of Cdk,k+1,k+2 , this has a natural action of (k + 1, k + 2). Proposition The invariants and anti-invariants of (k + 1, k + 2) are both free modules of rank 1 over Cdk,k+2 ∼ = H ∗ (Gr(k ⊂ k + 2, S)). Thus, my bimodule maps as a ring are a 2 × 2 matrix algebra over Cdk,k+2 . Definition Let the Demazure operator Di : Cd → Cd be the map Di (f ) =

f (z1 , . . . , zi+1 , zi , . . . , zd ) − f (z1 , . . . , zd ) zi+1 − zi

This kills polynomials invariant under (i, i + 1), and thus is well-defined. Ben Webster (UVA)

HRT : Lecture II

February 10, 2014

14 / 35

Demazure operators

Geometrically, I can think of the Demazure operator as integration along the projection map Gr(k ⊂ k + 1 ⊂ k + 2, S) → Gr(k ⊂ k + 2, S). Important facts: Di (f ) = 0 if f = f (i,i+1) Di (fg) = Di (f )g + f (i,i+1) Di (g). Thus, Di is a module map for Cdi−1,i+1 . Di (fg) = fDi (g) if f = f (i,i+1) Di (xi f ) − xi+1 Di (f ) = xi D(f ) − Di (xi+1 f ) = f , since Di (xi ) = 1, Di (xi+1 ) = −1. Di is the unique map satisfying these properties.

Ben Webster (UVA)

HRT : Lecture II

February 10, 2014

15 / 35

Demazure operators Theorem Multiplication by Cdk,k+1,k+2 and the Demazure operator Dk+1 generate k,k+1,k+2 bimodule endomorphisms of k+2 Hk+1 ⊗ k+1 Hk ∼ . = Cd In terms of pictures, I’ll draw a Demazure operator as a crossing: F

My relations then become: − F

F

F

= F

F

F

F

Theorem These relations give a presentation of End(k+2 Hk+1 ⊗ k+1 Hk ). Ben Webster (UVA)

HRT : Lecture II

February 10, 2014

16 / 35

Demazure operators Theorem Multiplication by Cdk,k+1,k+2 and the Demazure operator Dk+1 generate k,k+1,k+2 bimodule endomorphisms of k+2 Hk+1 ⊗ k+1 Hk ∼ . = Cd In terms of pictures, I’ll draw a Demazure operator as a crossing: F

My relations then become: − F

F

F

= F

F

F

F

Theorem These relations give a presentation of End(k+2 Hk+1 ⊗ k+1 Hk ). Ben Webster (UVA)

HRT : Lecture II

February 10, 2014

16 / 35

Demazure operators OK, let’s keep going; what about the triple tensor product? This is the endomorphisms of the module Cdk,k+1,k+2,k+3 . Here, the Demazure operators Dk+1 and Dk+2 both commute with Cdk,k+3 , and satisfy the relation: Dk+1 Dk+2 Dk+1 (f ) = Dk+2 Dk+1 Dk+2 (f ) = F

F

F

F

F

F

Theorem The endomorphisms of the functor Fk+` · · · Fk are generated by the Demazure operators and multiplication by elements of Cdk,k+1,...,k+` ; the relations these satsify are the ones we know. Ben Webster (UVA)

HRT : Lecture II

February 10, 2014

17 / 35

The nilHecke algebra Thus, on Fk+` · · · Fk , there’s a natural algebra that acts: Definition The nilHecke algebra N` is the algebra generated by symbols ψ1 , . . . , ψ`−1 , y1 , . . . , y` subject to the relations: ψi yi − yi+1 ψi = yi ψi − ψi yi+1 = 1 ψi ψj = ψj ψi ···

(|i − j| = 6 1)

···

yi yj = yj yi ···

yj

Ben Webster (UVA)

ψi ψi+1 ψi = ψi+1 ψi ψi+1

··· ψj

HRT : Lecture II

February 10, 2014

18 / 35

The nilHecke algebra The nilHecke algebra has a very natural geometric realization: let F`(`) be the `-step complete flag variety. We have 3 projections π1,2 , π1,3 , π2,3 : F`(`)3 → F`(`)2 . We can endow H∗GL` (F`(`) × F`(`)) with a convolution multiplication by ∗ ∗ A ? B = (π1,3 )∗ (π1,2 A ∩ π2,3 B).

Proposition We have a canonical isomorphism between the nilHecke algebra and H∗GL` (F`(`) × F`(`)) endowed with the convolution multiplication. The polynomial generators yi come from the equivariant homology of F`(`)∆ ; the classes ψi correspond to the fiber product F`(`) ×Gr(1⊂···⊂i−1⊂i+1⊂···⊂`) F`(`) ⊂ F`(`) × F`(`). Ben Webster (UVA)

HRT : Lecture II

February 10, 2014

19 / 35

The nilHecke algebra The other way to think about the nilHecke algebra N` is its natural representation on k[z1 , . . . , z` ] via multiplication and Demazure operators yi · f = zi f

ψi · f = Di (f ).

Theorem This action induces an isomorphism N` ∼ = Endk[z1 ,...,z` ]S` (k[z1 , . . . , z` ]) ∼ = M`! (k[z1 , . . . , z` ]S` ). In particular, N` and k[z1 , . . . , z` ]S` are Morita equivalent, and N` has a unique simple module C` which is `!-dimensional. It’s useful for us to note that `−2 e` = ψ1 ψ2 ψ1 ψ3 ψ2 ψ1 · · · ψ` ψ`−1 · · · ψ1 y`−1 · · · y`−1 1 y2

is a primitive idempotent. Thus, for any N` module, M ∼ = (e` M)⊕`! . Ben Webster (UVA)

HRT : Lecture II

February 10, 2014

20 / 35

Symmetric groups Fix a field k, and consider the group algebras of the symmetric groups Sn over k. These P are related by induction and restriction functors. Let Xm = i