Cyclotomic quiver Hecke algebras III

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Cyclotomic quiver Hecke algebras III

The Ariki-Brundan-Kleshchev categorification theorem

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 III 2 / 26

Ariki-Brundan-Kleshchev categorification theorem

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

· · ·

The aim for this lecture is to explain and understand:

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)coincides with

{[P]|self dual projective indecomposable Rn^Λ-module, n≥ 0}

• The dual canonical basis ofL_A(Λ) coincides with {[D]|self dual irreducibleRn^Λ-modules, n≥0}

Ariki proved the ungraded analogue of this result in 1996, establishing and generalising the LLT conjecture. This result motivated Chuang-Rouquier’s sl₂-categorification paper and the introduction of quiver Hecke algebras

Multipartitions and dominance

A multipartition, or `-partition, of n is an ordered`-tuple of partitions λ= (λ⁽¹⁾|λ⁽²⁾|. . .|λ^(`))such that |λ|=|λ⁽¹⁾|+· · ·+|λ^(`)|=n Let P_n^Λ be the set of`-partitions of n

We identify `-partitions and their diagrams:

[λ] ={(l,r,c)|1≤l ≤`,1≤c ≤λ^(l)_r } For example, if λ= (3,1|2,2|∅|1²)then

∅

A node is any triple (l,r,c)in a diagram. The set{1, . . . , `} ×N² of nodes is totally ordered by the lexicographic order

The set P_n^Λ is a post under dominance, where ifλ,µ∈ P_n^Λ then λBµ if

l−1

X

k=1

|λ^(k)|+

i

X

j=1

λ^(l_j ⁾≥

l−1

X

k=1

|µ^(k)|+

i

X

j=1

µ^(l_j ⁾

Dominance corresponds to moving nodes in the diagrams up and to the left

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Addable and removable nodes

Recall from last lecture that we fixed integers κ₁, . . . , κ_` such that

#{1≤l ≤`|κ_l ≡ i (mod e)}= (h_i,Λ), for i ∈I

An addable nodeofλis a nodeB ∈/ λsuch that λ+A:=λ∪ {B} ∈ P_n+1^Λ A removable nodeof λis a node B ∈λwith λ−A :=λ\ {B} ∈ P_n−1^Λ A node (l,r,c)∈ {1,2, . . . , `} ×N² is an i-nodeif it has residue

i =κ_l +c−r +eZ∈ I =Z/eZ

Let Add_i(λ) andRem_i(λ)be the sets of addable and removable i-nodes Definition (Brundan-Kleshchev-Wang)

If A is an addable or removable i-node of µ define:

d^A(µ) = #{B ∈Addi(µ)|A>B} −#{B ∈ Remi(µ)|A> B} d_A(µ) = #{B ∈ Add_i(µ)|A <B} −#{B ∈Rem_i(µ)|A<B} d_i(µ) = #Addi(µ)−#Remi(µ)

Standard tableaux

A λ-tableauis a mapt: [λ]−→ {1,2, . . . ,n}, which we identify with a labelling of [λ]. A tableau t isstandard if its entries increase along rows and down columns in each component

Let Std(λ) be the set of standard λ-tableaux

Example Let λ= (3,2|2²|(2,1). Then two standard λ-tableau are:

t^λ =

1 2 3 4 5

6 7 8 9

10 11 12

t_λ =

8 10 12 9 11

4 6 5 7

1 3 2

These are the initial and finalλ-tableau, respectively If t∈Std(λ) define permutations d(t)and d⁰(t)∈Sn by

t^λd(t) =t=t_λd⁰(t)

Theresidue sequenceoft∈Std(P_n^Λ)isi^t ∈Iⁿ where t⁻¹(m)is ani_m^t-node Let A=t⁻¹(n). Define thedegree and codegree, respectively, of t by:

degt= degt_↓(n−1)+d^A(µ) and deg⁰t= deg⁰t_↓(n−1)+d_A(µ) such that “empty(0|. . .|0)-tableau” has degree and codegree 0

By definition,degt,deg⁰t∈ Z. They can be positive, negative or zero

Cellular bases

The algebra Rn^Λ has two natural “dual” graded cellular bases.

For λ∈ P_n^Λ define polynomials y^λ=y(t^λ)and y_λ =y(t_λ) inductively by y^λ =y(t^λ_↓(n−1))y_n^d^A^(λ) and y_λ =y(t_λ_↓(n−1))y_n^d^A^(λ)

Then these two cellular bases have the following properties

Poset (P_n^Λ,D) (P_n^Λ,E)

Basis {ψ_st|(s,t)∈Std²(P_n^Λ)} {ψ_st⁰ |(s,t)∈ Std²(P_n^Λ)} Definition ψ_st =ψ^∗_d(s)i^λy^λψ_d(t) ψ_st =ψ_d^∗₀_(s)i_λy_λψ_d0(t)

Degree degψ_st = degs+ degt deg⁰ψ_st = deg⁰s+ deg⁰t

Residues i^s and i^t i^s and i^t

Cell modules S^λ S_λ

Simple modules D^µ Dµ

Let K_n^Λ= {µ∈ P_n^Λ|D^µ 6=0}andK_Λⁿ ={µ∈ P_n^Λ|D_µ 6= 0}. Then {D^µhki |µ∈ K^Λ_n,k ∈Z} and {D_µhki |µ∈ Kⁿ_Λ,k ∈Z} are both complete sets of pairwise non-isomorphic irreducible Rn^Λ-modules For symmetric groups, the Specht modules and simple modules are

interchanged by tensoring with the sign representation

Graded decomposition matrices

For λ∈ P_n^Λ and µ∈ K_n^Λ define graded decomposition numbers d_λµ(q) = [S^λ:D^µ]_q =P

k∈Z[S^λ :D^µhki]q^k ∈ N[q,q⁻¹] Let d_q = (d_λµ(q)) be the graded decomposition matrix

Let Y^µ be the (graded) projective cover of D^µ Let cq = ([Y^µ :D^ν]_q)_µ,ν∈KΛ

n be the graded Cartan matrix Theorem

Suppose that λ∈ P_n^Λ and µ∈ K_n^Λ. Then

d_µµ(q) =1andd_λµ(q)6= 0only if λDµ

Moreover, Y^µ has a filtration by graded Specht modules in which S^λ appears with multiplicityd_λµ(q)

=⇒ c_q = d^T_qd_q

Proof This follows from the general theory of (graded) cellular algebras Remark Specht filtration multiplicities are not well-defined, but the import of the theorem is that [Y^µ :S^λ]_q = [S^λ:D^µ]_q

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Induction and restriction

For i ∈I define 1n,i =P

j∈Iⁿ1ji ∈ Rn+1^Λ

Lemma

Let i ∈I. There is an embedding of graded algebrasRn^Λ ,→R_n+1^Λ given by 1j 7→1ji, y_r 7→y_r1n,i and ψ_s 7→ψ_s1n,i

This induces an exact functor

i-Ind :Rn^Λ-Mod−→Rn+1^Λ -Mod;M 7→M⊗_RΛ

n 1_n,iRn+1^Λ

Moreover, Ind=L

i∈Ii-Ind

Proof Check the KLR relations and use the KLR basis theorems The functor i-Ind has a natural left adjoint:

i-ResM =Me_1,i ∼= Hom_R^Λ

n(1n,iRn^Λ,M) Theorem (Kashiwara)

Suppose i ∈I. Then (i-Res,i-Ind) is a biadjoint pair.

Induction and restriction of Specht modules

Theorem (Brundan-Kleshchev-Wang, Hu-Mathas) Suppose that kis an integral domain and λ∈ P_n^Λ.

1 Let B₁>B₂>· · ·>B_y be the removablei-nodes of λ.

Then i-ResS^λ and i-ResS_λ have graded Specht filtrations 0=R0 ⊂R1⊂ · · · ⊂R_y =i-ResS^λ

0=R_y₊₁⊂ R_y ⊂ · · · ⊂R₁ =i-ResS_λ

such that R_j/R_j−1 ∼=q^d^Bj^(λ)S^λ−B^j andR_j/R_j+1∼=q^d^Bj^(λ)S_λ−B_j

2 Let A₁>A₂· · ·>A_z be the addable i-nodes ofλ.

Then i-IndS^λ andi-IndS_λ have graded Specht filtrations 0=I_z+1 ⊂I_z ⊂ · · · ⊂I₁=i-IndS^λ

0=I₀ ⊂I₁⊂ · · · ⊂ I_z =i-IndS_λ

such that I_j/I_j+1∼=q^d^Aj^(λ)S^λ+A^j andI_j/I_j−1∼= q^d^Aj^(λ)S_λ+A_j

Proof Reduce to the semisimple case and then use the seminormal form

Graded branching rules and tableaux degrees

0 1

0 −1 0 0

0 1 00 1 2 0 1

=⇒ [ResS^(3,1)] =q[S⁽³⁾] + [S^(2,1)] and [IndS₍₁3)] = [S_(2,12)] +q[S₍₁4)] (Brundan, Kleshchev and Wang) (Hu and M.) Paths still index a basis =⇒ dim_qS_(3,1)= q+q⁻¹+q

=⇒ dim_qD_(3,1) =q+q⁻¹

Defect and duality

Let ∗be the unique (homogeneous) anti-isomorphism of Rn^Λ that fixes each of the KLR generators

=⇒ (ψ_st)^∗= ψ_ts and(ψ_st⁰ )^∗=ψ⁰_ts, so∗is the cellular basis involution for both the ψ andψ⁰-bases

If M is an Rn^Λ-module then M^~ = Hom_k(M,k)is an Rn^Λ-module with action: (h·f)(m) =f(h^∗m), for h∈Rn^Λ,f ∈ M^~ and m∈M

=⇒ dimqM^~ = dimqM

where f(q) =f(q⁻¹) is theZ-linearbar involution on Z[q,q⁻¹] Previously, we noted that(D^µ)^~∼=D^µ and(D_ν)^~ ∼=D_ν

To describe duality on the Specht modules define the defectof β∈ Q⁺ defβ = (Λ, β)− ¹₂(β, β) = ¹₂

(Λ,Λ)−(Λ−β,Λ−β)

∈ N For λ∈ P_n^Λ set β_λ =Pn

k=1α_i^t

k ∈ Q⁺, for any t∈ Std(λ).

The defectof λisdefλ= defβ_λ

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Symmetrizing form

A graded k-algebra A is agraded symmetric algebra if there exists a homogeneous non-degenerate trace form τ:A−→k, where kis in degree zero. That is, τ(ab) =τ(ba)and if 06=a∈ A then there exists b∈A such thatτ(ab)6=0.

Theorem (Hu-M., Kang-Kasiwara, Webster)

Suppose that β ∈Q_n⁺. Then R_β^Λ a graded symmetric algebra with homogeneous trace form τ_β of degree −2defβ.

Proof Our proof reduces to the trace-form on Hn^Λ. A key part of the argument is the observation that

τ_β(ψstψ⁰_uv)6=0 only if uDt and that τ_β(ψstψ_ts⁰ )6=0 Corollary (Hu-M.)

Let λ∈ P_n^Λ. Then S^λ ∼=q^def^λS_λ^~ and S_λ =q^def^λ(S^λ)^~

Proof By the remarks above, an isomorphism is given by sending ψ_t ∈S^λ to the map θt∈q^def^λS_λ^~ that is given by θt(ψ_u⁰) =τ_β(ψ_tλtψ⁰_utλ)

The Hom-dual

Define # to be the graded endofunctor ofRep(Rn^Λ)andProj(Rn^Λ)given by M^#= Hom_RΛ

n(M,Rn^Λ)

In particular, note that (Y^µ)^# ∼=Y^µ since Y^µ is a summand of Rn^Λ

A straightforward argument using the adjointness of ⊗ and Hom gives:

Lemma

Let β∈Q⁺. As endofunctors of Rep(R_β^Λ), there is an isomorphism of functors #∼=q²^def^β◦~.

We use will ~ as the duality for the dual canonical bases and # for the canonical basis

The quantum group U

_q

( sl b

_e

)

Given our choice of Cartan matrix, we need to work with Uq(bsle) The quantum group Uq(slbe) associated with the Cartan matrixC is the Q(q)-algebra generated by {E_i,F_i,K_i^±|i ∈ I}, subject to the relations:

K_iK_j =K_jK_i, K_iK_i⁻¹=1, K_iE_jK_i⁻¹=q^c^ijE_j K_iF_jK_i⁻¹=q^−c^ijF_j, [E_i,F_j] =δ_ij^Kⁱ^−K

−1 i

q−q⁻¹ , P

0≤c≤1−cij(−1)^cq1−cij

c

y

iE_i^1−c^ij^−cE_jE_i^c =0 P

0≤c≤1−cij(−1)^cq1−cij

c

y

iF_i^1−c^ij^−cF_jF_i^c =0 where qd

c

y

i = ^J^d^Kⁱ^!

JcKⁱ!Jd−cKⁱ! and JmKⁱ! =Qm k=1

q^k−q^−k q−q⁻¹

Recall that A=Z[q,q⁻¹]

Let U_A(bsl_e)be Lusztig’s A-form ofU_q(bsl_e), which is theA-subalgebra of Uq(bsle)generated by the quantised divided powers

E_i^(k) =E_i^k/JkKⁱ! and F_i^(k) =F_i^k/JkKⁱ! For each Λ∈P⁺ there is a irreducible integrable highest weight module L(Λ) of highest weight Λ.

The combinatorial Fock space

The combinatorial Fock spaceF_A^Λ is the free A-module with basis the set of symbols { |λi |λ∈ P^Λ}, where P^Λ =S

n≥0P_n^Λ. For future use, let K^Λ =S

n≥0K^Λ_n . Set F_Q^Λ_(q)= F_A^Λ ⊗_AQ(q). Then,F_Q^Λ_(q) is an infinite dimensional Q(q)-vector space. We consider { |λi |λ∈ P^Λ}as a basis ofF_Q^Λ_(q) by identifying|λiand |λi ⊗1_Q_(q).

Theorem (Hayashi, Misra-Miwa)

Suppose that Λ∈P⁺. Then F_Q^Λ_(q) is an integrable U_q(bsl_e)-module with U_q(bsle)-action determined by

E_i|λi= X

B∈Rem_i(λ)

q^d^B^(λ)|λ−Bi, F_i|λi= X

A∈Add_i(λ)

q^−d^A^(λ)|λ+Ai

and K_i|λi=q^dⁱ^(λ)|λi, fori ∈I andλ∈ P_n^Λ. Proof A tedious check of the relations

It follows from the theorem that L(Λ)∼=Uq(bsle)|0_`i, where0_` is the zero `-partition. Define L_A(Λ) =U_A(bsl_e)|0_`i

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The CDE triangle in the Fock space

Recall that Rep(R^Λ) =L

n≥0Rep(Rn^Λ)and Proj(R^Λ) =L

n≥0Proj(Rn^Λ) Proposition

Suppose that Λ∈P⁺. Then thei-induction andi-restriction functors induce a U_q(bsl_e)-module structure on Proj(R^Λ)⊗_AQ(q) and

Rep(R^Λ)⊗_AQ(q)so that, as Uq(slbe)-modules,

Proj(R^Λ)⊗_AQ(q)∼=L(Λ)∼=Rep(R^Λ)⊗_AQ(q) Proof The decomposition matrix defines

the linear maps shown. As vector space homomorphisms, d^T_q is injective and d_q is surjective. Using the graded induction and restriction formulas it remains to observe that E_i coincides with i-Res and that q⁻¹F_iK_i coincides with i-Ind.

Proj(R^Λ) F_Q^Λ_(q)

Rep(R^Λ) d^T_q

dq

c_q

The result then follows since L(Λ) =U_q(bsle)v_Λ ⊆imd^T_q

Cartan pairing

A semilinearmap of A-modules is a Z-linear map θ:M−→N such that θ(f(q)m) =f(q)θ(m), for allf(q)∈A andm∈M.

A sesquilinear mapf :M×N−→A, where M and N are A-modules, is a function that is semilinear in the first variable and linear in the second.

Define theCartan pairing h[P],[M]i= dim_qHom_RΛ

n(P,M), for P ∈Proj(Rn^Λ)andM ∈Rep(Rn^Λ). This is a sesquilinear form because

Hom_RΛ

n(Phki,M)∼= Hom_RΛ

n(P,Mh−ki)

=⇒ h[Y^µ],[D^ν]i=δ_µν

The biadjointness of(E_i,F_i) implies that

hi-Indx,yi=hx,i-Resyi andhi-Resx,yi=hx,i-Indyi Using the uniqueness of the Shapovalov form, we obtain:

=⇒ If x ∈Proj(R^Λ)and y ∈Rep(R^Λ) then hd^T_q(x),yi=hx,d_q(y)i Corollary

As U_A(slb_e)-modules,Proj(R^Λ) =L_A(Λ)and

Rep(R^Λ) =L_A(Λ)^∨ ={x ∈L_Q_(q)(Λ)| hx,yi ∈Afor all y ∈LA(Λ)}

Dualities on Fock space

The dualities ~ and # on Rep(R^Λ)induce semilinear endomorphisms on Rep(R^Λ) andProj(R^Λ) by

[M]^~= [M^~] and [M]^# = [M^#] We concentrate on ~. Writed⁻¹_q = (e_µν(−q)) Lemma

Let λ∈ K^Λ_n. Then [S^µ]^~ = [S^µ] + X

µBτ∈K^Λ_n

a_µτ(q)[S^τ] Proof We just compute using the decomposition matrix:

[S^µ]^~= X

µBν∈K^Λ_n

d_µν(q)[D^ν]~

= X

µDν

d_µν(q) [D^ν]

= [S^µ] + X

τ∈K^Λ_n µBτ

X

ν∈K^Λ_n µDνDτ

d_µν(q)e_ντ(−q) [S^τ]

Lusztig’s Lemma

Proposition (Lusztig’s lemma)

There exists a unique basis {B^µ|µ∈ K^Λ}ofRep(R^Λ) such that (B^µ)^~=B^µ and B^µ = [S^µ] + X

µBτ∈K^Λ_n

b^µτ(q)[S^τ] where b^µτ(q)∈ δ_µτ +qZ[q]

Proof

Uniqueness IfB^µ and B˙^µ are two such elements then B^µ−B˙^µ =P

µBτ c^µτ(q)[S^τ], for c^µτ(q)∈qZ[q].

The left-hand side is ~-invariant and c^µτ(q)∈ q⁻¹Z[q⁻¹]. If τ 6=µ is maximal such that c^µτ(q)6= 0then the last lemma forces

c^µτ(q)∈qZ[q]∩q⁻¹Z[q⁻¹] ={0}, a contradiction! Hence,B^µ = ˙B^µ

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Lusztig’s lemma – existence

Existence: argue by induction on dominance

If µ∈ K_n^Λ is minimal in K_n^Λ then B^µ = [S^µ] = [D^µ] = (B^µ)^~. If µ∈ K_n^Λ is not minimal set C^µ = [D^µ]

=⇒ (C^µ)^~=C^µ and C^µ = [S^µ] +P

µBτc^µτ(q)[S^τ], for c^µτ(q)∈A

If c^µτ(q)∈qZ[q] for all τ, set B^µ = C^µ – we’re done If not, let ν be maximal such that c^µν(q)∈/ qZ[q]

Replace C^µ with the element C^µ−a^µν(q)B^ν, wherea^µν(q)is the unique Laurent polynomial such that a^µν(q) =a^µν(q) and

c^µν(q)−a^µν(q)∈ qZ[q].

=⇒ (C^µ)^~=C^µ and the coefficient of [S^ν] inC^µ belongs to qZ[q].

Repeating this process, after finitely many steps we construct an

element B^µ with the required properties. 2

Canonical basis

Using an almost identical argument starting with Xµ =P

λ∈K^Λ_n e_λµ(−q)[Y^λ]∈Proj(R^Λ) we obtain:

Proposition (Lusztig’s lemma)

There exists a unique basis {B_µ|µ∈ K^Λ}of Proj(R^Λ) such that (B^µ)^# =B^µ and B^µ = [S_µ] + X

τBµ∈K^Λ_n

b^{τ µ}(q)[X_τ] where b^{τ µ}(q)∈ δ_{τ µ}+qZ[q]

The basis {B^µ} is thedual canonical basis of L_A(λ)^∨∼=Rep(R^Λ)and {D_µ|µ∈ K^Λ}is the canonical basisof L_A(Λ)∼=Proj(Rn^Λ)

As their names suggest, these two bases are dual under the Cartan pairing:

Corollary

Suppose that λ,µ∈ K^Λ. Then hB_µ,B^λi=δ_λµ

Ariki’s categorification theorem

Let Proj(H^Λ) =L

n≥0Proj(Hn^Λ) be the Grothendieck group of the ungraded algebras Hn^Λ, forn≥ 0.

=⇒ Proj(H^Λ) is the freeZ-module with basis{Y^µ|µ∈ K^Λ}, where M 7→M is the forgetful functor that forgets the grading Let L₁(Λ) be the irreducible integrable highest weight module with highest weight Λwhenq =1

Theorem (Ariki’s Categorification Theorem)

Suppose that k is a field of characteristic zero. Then the canonical basis of L₁(Λ)coincides with the basis of (ungraded)projective indecomposable Hn^Λ-modules {[Y^µ]|µ∈ K^Λ}of Proj(Hn^Λ).

Corollary

Suppose that k is a field of characteristic zero. Then {[D^µ]|µ∈ K^Λ}is the dual canonical basis of L_A(Λ)

=⇒ d_λµ(q)∈δ_λµ+qN[q]

Cyclotomic quiver Hecke algebras III