Cyclotomic quiver Hecke algebras IV

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Cyclotomic quiver Hecke algebras IV

Applications and other types

Andrew Mathas

University of Sydney

Categorification, representation theory and symplectic geometry Hausdorff Research Institute for Mathematics

November 2017

Outline of lectures

1 Quiver Hecke algebras and categorification Basis theorems for quiver Hecke algebras Categorification ofU_q(g)

Categorification of highest weight modules

2 The Brundan-Kleshchev graded isomorphism theorem Seminormal forms and semisimple KLR algebras Lifting idempotents

Cellular algebras

3 The Ariki-Brundan-Kleshchev categorification theorem Dual cell modules

Graded induction and restriction The categorification theorem

4 Recent developments

Consequences of the categorification theorem Webster diagrams and tableaux

Content systems and seminormal forms

Andrew Mathas—Cyclotomic quiver Hecke algebras IV 2 / 28

Ariki-Brundan-Kleshchev categorification theorem

Let C be a generalised Cartan matrix of type A⁽¹⁾_e or A_∞:

· · · Last lecture we saw that

Theorem (Ariki, Brundan-Kleshchev, Brundan-Stroppel, Rouquier)

Let C be a Cartan matrix of type A⁽¹⁾e or A_∞ and let kbe a field. Then L_A(Λ)∼= M

n≥0

Proj(Rn^Λ) and L_A(Λ)^∨∼= M

n≥0

Rep(Rn^Λ)

Moreover, if k=C then

• The canonical basis ofL_A(Λ) is{[Y^µ]|µ∈ K^Λ}

• The dual canonical basis ofL_A(Λ)is {[D^µ]|µ∈ K^Λ}

If µ∈ K^Λ then (D^µ)^~∼=D^µ and (Y^µ)^#∼= Y^µ, where if M is an Rn^Λ-module thenM^~= Hom_k(M,k)and M^# = Hom_RΛ

n(M,Rn^Λ)

Categorification of the canonical basis of U

_A

(b sl

_e

)

⁺

Set Proj(R) =L

n≥0Proj(Rn) and let# be the automorphism of Proj(R) induced by M^# = Hom_R_n(MRn)

Theorem (Brundan-Kleshchev, Brundan-Stroppel, Rouquier) Let C be a Cartan matrix of type A⁽¹⁾_e or A_∞ and let k=C.

ThenU_A⁻(bsl_e)∼=Proj(R) and the canonical basis ofU_A(bsl_e) coincides with the basis of Proj(R) of#-self-dual projective indecomposable Rn-modules Proof Let Bbe the canonical basis of U_A⁻(bsle)and B_Λ be the canonical basis ofLA(Λ) =U_A(bsle)v_Λ, for Λ∈P⁺. ThenB is the uniqueweight basis ofU_A⁻(bsl_e)such that ifb ∈Bthen bv_Λ ∈B_Λ∪ {0}

As B_Λ ={[Y^µ]|µ∈ K^Λ_n}, it is enough to show that if Y is a self-dual Rn-module then [Y]v_Λ is either zero or equal to[Y^µ], for someµ∈ K^Λ Define a functorpr_Λ:Rn-Mod−→Rn^Λ-Mod by pr_ΛM =Rn^Λ⊗_R_n M

=⇒ pr_Λ sends projectives to projectives and pr_Λ◦#∼= #◦pr_Λ This implies the result

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Simple modules

By definition, K^Λ_n ={µ∈ P_n^Λ|D^µ 6=0}but we did not describe this set Given i-nodesA<C, for i ∈ I, define

d_A^C(µ) = #{B ∈Addi(µ)|A<B<C} −#{B ∈ Remi(µ)|A<B<C} A removable i-node A isnormalif d_A(µ)≤ 0andd_A^C(µ)<0whenever C ∈ Remi(µ)and A<C.

A normal i-nodeA is good if A≤B wheneverB is a normali-node.

Write λ−−−−−ⁱ^-good→µ if µ =λ+A for some good i-node A.

Misra and Miwa showed that the crystal graph of L_A(Λ), considered as a submodule FA^Λ, is the graph with vertex set

L0^Λ= {µ∈ P^Λ|µ=0_` orλ−−−−−ⁱ^-good→µ for some λ∈ L0^Λ}, and labelled edges λ−−−−−^i-good→µ, for i ∈I

Corollary (Ariki)

Suppose that k is an arbitrary field and thatµ∈ P_n^Λ. Then K^Λ =L0^Λ. That is, if µ∈ P_n^Λ then D_k^µ 6=0 if and only ifµ∈L₀^Λ

Proof Immediate because [D^µ] = [S^µ]+lower terms, for µ∈ K^Λ

Categorification of highest weight modules

The categorification of L(Λ)^∨ and L(Λ) by the algebras Rn^Λ is extensive:

•Multiplication by q corresponds to the grading shift functor

•E_i ↔i-Res and F_i ↔q i-IndK_i⁻¹

•The weight spaces of L(Λ)are the blocks of Rn^Λ

•The Shapovalov form on L(Λ) is the Cartan pairing on Rep(Rn^Λ)

•The standard basis of L(Λ) corresponds to the graded Specht modules

•The costandard basis of L(Λ) corresponds to the dual graded Specht modules

•The vertices of the crystal graph label the simple modules

•The crystal graph gives the modular branching rules

•The action of the affine Weyl group corresponds to the derived equivalences of Chuang and Rouquier

•If F =C the dual canonical basis is the basis of irreducible modules

•If F =C the canonical basis is the basis of projective indecomposable modules

Almost simple modules

The quiver Hecke algebra Rn^Λ(Z)is defined overZ (but Rn^Λ(Z)6∼=Hn^Λ(Z)!) Rn^Λ(Z) is aZ-graded Z-free cellular algebra

=⇒ S^λ

Z is defined overZ with aZ-valued bilinear form h, i Define radS^λ

Z ={x ∈S^λ

Z | hx,yi=0 for all y ∈S^λ

Z }

=⇒ radS^λ

Z is a Z-graded Z-free submodule of S^λ

Z

Definition Let E^µ

Z =S^µ

Z/radS^µ

Z. For a fieldk, let E_k^µ = E^µ

Z ⊗_Zk, an Rn^Λ(k)-module Theorem (M., Brundan-Kleshchev)

1 The moduleE^µ

Z is aZ-graded Z-free Rn^Λ(Z)-module

2 If k=Q then E^µ

Q is self-dual and, moreover,E^µ

Q

∼=D^µ

Q is an absolutely irreducible gradedRn^Λ(Q)-module

3 For anyλ∈ P_n^Λ and µ∈ K^Λ_n, [S_k^λ :D_k^µ]_q =X

ν

[S_Q^λ :E_Q^ν]_q[E_k^ν :D_k^µ]_q = X

ν

d_λν(q) [E_k^ν :D_k^µ]_q adjustment matrix

The James conjecture

James conjecture (1990)

Let kbe a field of characteristic p and α∈Q⁺ such that defα <p. Then theadjustment matrixfor Rα^Λ⁰ ∼=Hξ(S_n)_α is the identity matrix.

Proving the James and Lusztig conjectures motivated developments in representation theory for the last twenty years.

Evidence for James and Lusztig conjectures

•(Andersen-Jantzen-Soergel) True for almost all primes

•True for n≤30 (James, M., ...)

•True for blocks of defect/weight 1, 2 (Richards) and 3 and 4 (Fayers)

•True for theRouquier Blocks, which have arbitrary weight (Chuang-Tan, James-Lyle-M.)

Williamson (2013)

The James and Lusztig conjectures are both wrong!!!

The smallest known counter-example to the James conjecture occurs in a block of defect 561in characteristic 839for the symmetric group S467,874

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Loadings and Webster tableaux

Recall that Λ∈P⁺ is a dominant weight of level`.

A loadingis a sequence θ= (θ₁, . . . , θ_`)∈ Z^` such that

θ1< θ2<· · ·< θ_` and θ_k 6≡θ_l (mod `) for1≤k <l ≤ ` Extend θ to the set of nodes by defining

θ(l,r,c) =Nθ_l +L(c−r) +r +c−1

where L=N`and N (2n−1) — different nodes have different loadings The loading ofλ∈ P_n^Λ isL_θ(λ) ={θ(α)|α∈[λ]}

Define the θ-dominance order on P_n^Λ by λBθ µ if for all nodes(l,r,c)

#{α∈ [λ]|θ(α)≥ θ(l,s,d)} ≥#{(α)∈[µ]|θ(α)≥ θ(l,s,d)} A Websterλ-tableauof type µ is a bijection T: [λ]−→L_θ(µ)such that

1 If 1≤k ≤` andλ^(k) 6= (0) then T(k,1,1)≤Nθ_k

2 If (k,r−1,c),(k,r,c)∈ λthen T(k,r −1,c)<T(k,r,c) +L

3 If (k,r,c−1),(k,r,c)∈ λthen T(k,r,c−1)<T(k,r,c)−L Let SStdθ(λ,µ)be the set of Websterλ-tableau of type µ and let SStd_θ(λ) =S

µSStd_θ(λ,µ). Let ω_n = (0|. . .|0|1ⁿ).

Then Stdθ(λ) =SStdθ(λ,ω_n)is the set of standard Webster tableau

Many cellular bases

Theorem (Bowman, Webster, 2017)

Let θ be a loading. ThenRn^Λ has a graded cellular basis {c_st^θ |s,t∈Stdθ(λ),λ∈ P_n^Λ} with respect to the poset(P_n^Λ,Bθ)

Let C_λ^θ be the cell module indexed by λ∈ P_n^Λ determined by the θ-cellular basis {c_st^θ } and let D_µ^θ =C_µ^θ/radC_µ^θ. Define

d_λµ^θ (q) = [C_λ^θ :D_µ^θ]_q =X

k∈Z

[C_λ^θ :D_µ^θhki]q^k

The θ-cellular bases genuinely depend on θ and they are in general different from the ψ andψ⁰-bases. In fact, the graded dimensions of the θ-cell modules and the graded decomposition numbers depend on θ

Graded decomposition example

Example (Bowman) Take e= 2, n=2, Λ = Λ0+ Λ1 andκ= (0,1). Then R2^Λ has two one dimensional self-dual simple modules,D(01)and D(10), such that 1i acts as δ_ij on D(j)

θ= (0,1)

D(10) D(01) (1²|0) 1

(0|1²) · 1

(1|1) q q

(2|0) q² ·

(0|2) · q²

θ= (0,3)

D(10) D(01) (0|1²) 1

(0|2) q

(1|1) q² 1

(1²|0) · q

(2|0) · q²

Before we can define the {c_st^θ }basis we need to introduce a new algebra

Webster tableau

For anyλ there is always a unique Websterλ-tableau of type λ. Rather than drawing this “normally” we want to draw Webster tableau using the Russian notation

Example Takeλ= (2²|2,1), so thatN =15 andL= 30, for the two loadingsθ = (0,1) andθ= (0,5), respectively:

0 15 x 1 -28 32

3

16 -13 47

0 75 x

1 -28 32

3

76 47 107

θ= (0,1) θ= (0,5)

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Webster diagrams

The elements of Rn can be described diagrammatically:

1_i =

i1i2 ir in

y_r1_i =

i1i2 ir in

ψ_r1_i =

i1i2 ir+1ir in

We want similar, but more complicated diagrams, to define an algebra Wn^Λ

Webster diagrams have three types of strings:

• Thick red vertical strings withx-coordinates Nθ1, . . . ,Nθ_`

• Solid strings of residues i₁, . . . ,i_n, for some i∈Iⁿ

• Dashed greyghost strings that are translates,L-units to the left, of the solid strings. A ghost string has the

same residue as the corresponding solid string

Diagrams are defined up to isotopy and solid strings can have dots The following crossings are notallowed for red, solid or ghost strings):

σ₁

σ₂

Examples of Webster diagrams

Example Let `=1, θ= (0)andλ= (4,2,1). Then N =15=L and SStd_θ(λ, λ)contains the tableau:

1 -13 -27 -42

17 3 33

1 2 3 4

5

6 7

The corresponding Webster diagram1_λ is:

1 2

3

4 6 5 7

Examples of Webster diagrams II

Now let λ= (2²|2,1), so thatN =15 andL=30.

If θ= (0,1) then 1ⁱ_λ is the diagram

1

2 6 4 5 3 7

If θ= (0,5) then 1ⁱ_λ looks like:

1

2 4 3 6 5 7

Strings in diagrams from `-partitions “cluster” according to the diagonals:

ρ₋₁solid strings

−1th diagonal

ρ0solid strings 0th diagonal

ρ1solid strings 1st diagonal

Composing Webster diagrams

We compose Webster diagrams in the usual way: if D and E are Webster diagrams then the diagram D◦E is0if their residues are different and when their residues are the same we put D on top of E and apply isotopy.

For example ifD is the diagram

Let E be the diagram obtained by reflectingD in the line y =0. Then D◦E is the diagram

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Relations for Webster algebras

The Webster algebra Wn^Λ is the k-algebra spanned by isotopy classes of Webster diagrams with multiplication given composition and subject to the following local relations:

i j

−

i j

= δ_ij

i j

=

i j

−

i j

=

i j

and

i j

=

i j

= and =

i j

=δ_ij

i j

and

k i j

=

k i j

Relations for Webster algebras II

i_ri_s i_s

=q_i_r_i_s(y_r,y_s)

i_si_s i_s

and

i_si_r i_s

=q_i_r_i_s(y_r,y_s)

i_si_r i_s

ir+1

i_r i_s i_r i_r+1

=

ir+1

i_r i_s i_r i_r+1

+ δ_rq_i_r_i_r+1(y_r,y_r+1)

i_r i_sir+1 i_r i_r

i_r+1 i_r i_s i_s

=

i_r+1 i_r i_s i_s

+δ_rq_i_r_i_r₊₁(y_r,y_r+1)

i_r i_si_r+1 i_s

= and =

i j

=

i j and

i j

=

i j

if i 6=j

Relations for Webster algebras III

i i

=

i i and

ρ_r i

= κ_i(y_r)

i i

j k i

=

i j k

+ δ_ijδ_jk

i j k

= and

=

... and a solid strand in D isunsteadyif it intersects the region (∞,LN]×[0,L], in which caseD =0

These relations are homogeneous, so Wn^Λ is a graded algebra

The cellular basis

Inside Wn^Λ, forT∈SStd_θ(λ,µ) define the diagram C_T to be a Webster diagram with a minimal number of crossings such that for each node (l,r,c)∈[λ]there is a solid string of residue κ_l +c−r +eZ that starts withx-coordinateT(l,r,c)∈ L_θ(µ) at the top of the diagram and that finishes withx-coordinate θ(l,r,c)∈ L_θ(λ) at the bottom of the diagram.

The diagram C_T is not unique, in general.

Let C_T^∗ be the diagram obtained fromCT by reflecting it in the line y =0.

Define C_ST^θ =C_SC_T^∗

Theorem (Bowman, Webster)

The algebraWn^Λ is spanned by the diagrams

{C_ST^θ |S,T∈SStdθ(λ)for λ∈ P_n^Λ}

Idea of proof First push all strings to the left so that they are concave, turning at the equator. This shows that if D is a Webster diagram then D ∈ Wn^Λ1_λWn^Λ, for someλ∈ P_n^Λ.

By resolving crossing it now follows that Wn^Λ is spanned by the {C_ST^θ }

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Connection to KLR

Recall that ω_n= (0|. . .|0|1ⁿ) and thatStdθ(λ) =SStdθ(λ,ω_n) Theorem (Bowman, Webster)

There is an isomorphism of graded algebras Rn^Λ

−'→1ωnWn^Λ1ωn

Idea of proof The isomorphism is given by:

1_i 7→

ρ₁ ρ_` i1 i_r i_n

y_r1i 7→

ρ1 ρ_` i₁ i_r i_n

ψ_r1i 7→

ρ₁ ρ_` i1 i_r ir+1 i_n

Pull all strings to the right and check the relations

The loaded cellular basis

For s,t∈Stdθ(λ) define c_st^θ to be the element of Rn^Λ that is sent to C_st^θ under the previous isomorphism.

Corollary (Bowman, Webtser)

The elements{c_st^θ |(s,t)∈Std²(P_n^Λ)}span Rn^Λ

Theorem (Bowman, Webster)

The algebraWn^Λ is quasi-heredity over k with graded cellular basis {C_ST^θ |S,T∈SStdθ(λ) forλ∈ P_n^Λ}

Bowman callsWn^Λ a diagrammatic Cherednik algebra. These algebras include, as a special case, the quiver Schur algebras of type A introduced by Stroppel and Webster (and Hu and Mathas in type A_∞). Webster proves thatWn^Λ categorifies Uglov’s generalised Fock spaces

The last theorem provides us with a quotient functor, or Schur functor:

E_ω_n:Wn^Λ-Mod−→Rn^Λ-Mod;M 7→1_ω_nM

Content systems

In the most general set up, the cyclotomic quiver Hecke algebra depends on choice of polynomials Q_I = Q_ij(u,v)

and K_I = κ_i(u)

, so write Rn^Λ =Rn^Λ(Q_I,K_I)

Fix ρ∈I^` such that Λ =P

lΛρ_l

Let Γ_` be the quiver of type A_∞× · · · ×A_∞, with ` factors.

More explicitly, Γ_` has vertex set J_` = [1, `]×Z and edges (l,a)−→(l,a+1), for all(l,a)∈ J_` Definition (Evseev-M.)

A content systemfor Rn^Λ(QI,KI) is a pair of maps r:J_`−→I and c:J_`−→ksuch that

• r(l,0) =ρ_l andκ_i(u) =Q

l∈[1,`],ρl≡i(u−c(l,0))

• Ifi =r(k,a)then Q_ij c(k,a),v 'Q

b v −c(k,b) , where b= a±1and r(k,b) =j

• If(k,a),(l,b)∈ J_` then r(k,a) =r(l,b)and c(k,a) =c(l,b) if and only if(k,a) = (l,b)

• Plus one more technical constraint

Examples of content systems

•If Γ =A_∞t · · · tA_∞, so thatI =J_`, then r(k,a) = (k,a) and c(k,a) =0 is a content system with coefficients in Z

•If Γ is a quiver of typeA⁽¹⁾_e+1 then a content system is given by:

r 0 1 2 . . . e 0 1 . . .

c 0 x 2x . . . ex (e+1)x (e+2)x . . .

•If Γ is a quiver of typeC_e⁽¹⁾ then

r 0 1 . . . e−1 e e−1 . . . 1 0 1 . . .

c 0 x . . . (e−1)x (ex)² −(e+1)x . . . −(2e−1)x (2x)² (2e+1)x . . .

Content systems are not unique – the most generic content systems are defined over Z[x,x₁, . . . ,x_`]

All of these content systems, and hence the algebras Rn^Λ(Q_I,K_I)are defined over Z[x]. There is a natural (homogeneous) specialisation map Rn^Λ(QI,KI)−→Rn^Λ)given by tensoring with Z[x]/xZ[x]

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Seminormal representations

Proposition

Let λ∈ P_n^Λ and let V be the K-vector space with a basis {v_t|t∈Std(λ)}and set v_s=0 if sis not standard.

Suppose that there exist scalars

{β_r(t)∈K|1≤r <n and t,s_rt∈Std(λ)} satisfying certain technical conditions.

Then V has the structure of an irreducible Rn^Λ(Q_I,K_I)-module where the Rn^Λ(QI,KI)-action is determined by:

1_ivt =δ_i,i^tvt

y_rv_t =c_r(t)v_t ψ_rv_t=β_r(t)v_s_r_t+

δ_it r,it

r+1

cr+1(t)−cr(t)v_t for all i ∈Iⁿ, all admissibler and allt∈Std(λ).

Idea of Proof Check the relations - the result comes from the normal machinery from seminormal forms

Semisimplicity and cellularity

Theorem (Evseev-M.)

Suppose that Rn^Λ(Q_I,K_I) has a content system over kand let K be the field of fractions of k. Then Rn^Λ(QI,KI) is a split semisimple graded K-algebra that is canonically isomorphic to a cyclotomic quiver Hecke algebra for the quiver A_∞t · · · tA_∞ with vertex set J_`.

The algebraRn^Λ(Q_I,K_I)has “integral” analogues of the ψ and ψ⁰-bases.

Unfortunately, it is not at all clear that these elements span the algebra.

Using a variation of the algebras Wn^Λ we can prove:

Theorem (Evseev-M.)

Suppose thatRn^Λ(QI,KI)has a content system over k. The Rn^Λ(QI,KI)is a graded cellular algebra

Corollary (Evseev-M.)

Let Rn^Λ be a quiver Hecke algebra of type C_e⁽¹⁾. Then Rn^Λ is a graded cellular algebra:ls

Cyclotomic quiver Hecke algebras IV