Cyclotomic quiver Hecke algebras II

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Cyclotomic quiver Hecke algebras II

The Graded Isomorphism 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 II 2 / 22

Quiver Hecke algebras of type A

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

· · ·

Fix Λ∈P⁺ and define Q-polynomials andκ-polynomials by:

Qij(u,v) =











(u−v)(v−u) ifij, u−v, ifi−→j

v−u, ifi←−j

1, ifi / j

0, ifi=j

and κi(u) =u^hhⁱ^,Λi

Then Rn^Λ =L

α∈Qn⁺Rα^Λ, where Rα^Λ is generated by {1_i|i∈I^α} ∪ {ψ_r|1≤r <n} ∪ {y_r|1≤r ≤n} with relations

κ_i₁(y₁)1i= 0, 1i1j= δ_i,j1i, P

i∈I^α1i =1, ψ_r1i =1_s_riψ_r, y_r1_i =1_iy_r, y_ry_t = y_ty_r, ψ_r²1_i =Q_i_r_,i_r₊₁(y_r,yr+1)1_i

ψryt =ytψr if s 6=r,r+1, ψrψt =ψtψr if |r −t|>1 (ψ_ry_r+1−y_rψ_r)1i =δ_i_r_,i_r₊₁1i = (y_r+1ψ_r −ψ_ry_r)1i

(ψr+1ψ_rψr+1−ψ_rψr+1ψ_r)1_i =∂Q_i_r_,i_r₊₁_,i_r+1(y_r,yr+1,yr+1)1_i

Cyclotomic Hecke algebras of type A

Fixξ∈ ksuch that e is minimal with 1+ξ²+· · ·+ξ^2(e−1)= 0 Fix integersκ1, . . . , κ_` such that for alli ∈I,

#{1≤l ≤`|κ_l ≡i (mod e)}= (h_i,Λ)

For m∈N and define theξ-quantum integer [m] = [m]_ξ = _ξ−ξ^ξ^2m⁻¹₋₁ Definition

The cyclotomic Hecke algebra of typeA is the unital associative k-algebra Hn^Λ= Hn^Λ(ξ) with generatorsT₁, . . . ,T_n−1, L₁, . . . ,L_n and relations

Q`

l=1(L₁−[κ_l]) =0, (T_r−ξ)(T_r +ξ⁻¹) =0, L_rL_t = L_tL_r T_sTs+1T_s =Ts+1T_sTs+1, T_rT_s =T_sT_r if |r −s|>1

T_rL_t =L_tT_r if t 6=r,r +1, L_r+1=T_rL_rT_r +T_r

When ξ²6=,Hn^Λ is an Ariki-Koike algebra, which is a deformation of the group algebra ofZ/`ZoS_n. If ξ²=1then Hn^Λ is adegenerate Ariki-Koike algebra. If `=1and ξ²=1 then Hn^Λ ∼=kS_n.

Theorem (Ariki-Koike) The algebra Hn^Λ is free as a k-module with basis {L^a₁¹. . . ,L^a_nⁿT_w|0≤a_k < ` and w ∈ S_n},

In particular,Hn^Λ is free of rank`ⁿn! = #(Z/`ZoS_n)

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The Brundan-Kleshchev graded isomorphism theorem

Theorem (Brundan-Kleshchev, Rouquier) Suppose that k is a field. Then Hn^Λ ∼=Rn^Λ. Remarks

This theorem is only true whenkis a field. For example, both algebras are defined over Z[ξ]but in general the theorem is false over this ring As a consequence,Hn^Λ is aZ-graded algebra

Brundan and Kleshchev prove this by constructing two explicit maps Rn^Λ −→Hn^Λ and Hn^Λ−→Rn^Λ and then checking the relations on both sides: nice result, ugly proof

The aim for today is to prove half of this theorem, concentrating on kS_n. At the same time, we will try to understand the KLR relations

Corollary

Suppose that k is a field and that ξ, ξ⁰∈ kare elements with e >1 minimal such that [e]_ξ =0= [e]_ξ0. ThenHn^Λ(ξ)∼=Hn^Λ(ξ⁰)

Jucys-Murphy elements and the Gelfand-Zetlin subalgebra

The presentation of Hn^Λ includes theJucys-Murphy elements L₁, . . . ,L_n In the case of the symmetric group (or their Iwahori-Hecke algebra),

L_k = (1,k) + (2,k) +· · ·+ (k −1,k) (an “averaging operator”) Definition

The Gelfand-Zetland subalgebra ofHn^Λ isLn^Λ =hL₁, . . . ,L_ni

Okounkov and Vershik have given a beautiful account of the semisimple representation theory ofS_n, by showing that

Ln^Λ= {z ∈kSn|zh=hz for all h∈kS_n−1}

They use Ln^Λ to show that the restriction of any irreducible CS_n-module is multiplicity free and from this deduce that every irreducible CSn-module has a basis of simultaneous eigenvectors for the elements ofLn^Λ and they deduce what the eigenvalues are.

Theorem

Let kbe a field. Then Hn^Λ is (split) semisimple if and only ifLn^Λ is (split) semisimple

Tableau combinatorics

A partitionof n is a weakly decreasing sequenceλ1 ≥λ2≥ · · · ≥0 of non-negative integers that sum to n. Identifyλ with its Young diagram [λ] ={(r,c)|1≤c ≤λ_r}, which is an array of boxes in the plane.

Let P_n^Λ be the set of partitions of n Example The diagram of (3,2)is

A λ-tableauis a function t: [λ]−→ {1,2, . . . ,n}, which we think of as a labelled diagram. A λ-tableau is standard if its entries increase along rows and down columns.

Let Std(λ)be the set of standard λ-tableaux andStd(P_n^Λ) =S

λ∈P_n^ΛStd(λ) Example The standard (3,2)-tableaux are:

1 2 3

4 5 , 1 2 4

3 5 , 1 2 5

3 4 , 1 3 4

2 5 , 1 3 5 2 4

Remark If ` >1then partitions get replaced by `-tuples of partitions and standard tableau get replaced by `-tuples of tableaux whose entries increase along rows and down columns in each component.

Content functions

The contentof a node(r,c)isc−r and ift is standard and1≤m≤n then thecontent of m int isc_m(t) =c−r, ift(r,c) =m

Example If λ= (4,3,3,2) then the contents in [λ]are:

0 1 2 3

−1 0 1

−2−1 0

−3−2

Contents increase along rows, decrease down columns and are constant on the diagonals ofλ. The addable nodes ofλ have distinct contents

Lemma

Let s∈ Std(λ)and t∈ Std(µ). Then s=tif and only if c_m(s) =c_m(t) for 1≤m≤n. Consequently, if 1≤r < nthen cm(t) =cm(t)for r 6= m,m+1 if and only ifs=tor s=s_rt

Proof Follows easily by induction because addable nodes have distinct contents

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Seminormal forms

Theorem (Young’s seminormal form, 1901)

Let λ be a partition. Define theSpecht module S^λ to be theQS_n-module with basis {v_t|t∈Std(λ)}and where the S_n-action is determined by

s_rv_t = _ρ¹

r(t)v_t+ ^ρ^r^(t)+1_ρ

r v_s_r_t,

where ρ_r(t) =cr+1(t)−c_r(t) andv_s_r_t=0if s_rt∈/ Std(λ)

Key point Lett∈Std(λ) and1≤m≤n. ThenL_mv_t= c_m(t)v_t (†) Assume only (†)and write s_rv_t= P

sa_stv_s If m6=r,r +1 then P

sc_m(s)a_stv_s= L_ms_rv_t=s_rL_mv_t =c_m(t)s_rv_t

=⇒ a_st 6=0 only ifs=t or s=s_rt

Let s=s_rt and write s_rv_t=αv_t+βv_s ands_rv_s =α⁰v_s+β⁰v_t

=⇒ (1) v_t= (α²+ββ⁰)v_t+ (α−α⁰)βv_s

=⇒ (2) αc_r(t)v_t +βcr+1(t)v_s =L_rs_rv_t = (s_rLr+1−1)v_t

=⇒ α= _c ¹

r+1(t)−cr(t) = _ρ¹

r(t) andββ⁰ =1− _ρ ¹

r(t)² = ^(ρ^r^(t)−1)(ρ_ρ ^r^(t)+1)

r(t)²

Young idempotents

For ta standard tableau define F_t=

n

Y

r=1

Y

sstandard cr(s)6=cr(t)

Lr−cr(s) cr(t)−cr(s)

Theorem

Suppose that kis a field of characteristic p>n. Then:

1 {F_t|ta standard tableau of size n}is a complete set of pairwise orthogonal idempotents

2 If λ∈ P_n^Λ and t ∈Std(λ) then S^λ∼= kS_nF_t

3 {S^λ|λ∈ P_n^Λ} is a complete set of pairwise non-isomorphic kS_n-modules

4 As an (Ln^Λ,Ln^Λ)-bimodule, kS_n =L

(kS_n)_st, where

(kS_n)_st= {a∈ kS_n|L_ra =c_r(s)a andaL_r =c_r(t)a} is one dimensional for all s,t∈Std(λ), λ∈ P_n^Λ

By part (4),kS_n has a basis {f_st|(s,t)∈Std²(P_n^Λ)}with f_st ∈(kS_n)_st

=⇒ f_stf_uv =δ_tvγ_tf_sv, for someγ_t∈ k =⇒ F_t= _γ¹

tf_tt

A nice action on seminormal bases

The action of kSn on the seminormal basis {f_st}is given by L_rf_st =c_r(s)f_st and s_rf_st = _ρ¹

r(s)f_st+β_r(s)f_ut, where u= s_rs As the L_r’s are acting by scalars they are essentially irrelevant. Indeed, the action of Ln^Λ on the seminormal basis is determined by F_vf_st = δ_svf_st We can “simplify” the action of s_r by defining

ψ_r = X

v∈Std(P_n^Λ) 1

βr(v)(s_r − _ρ¹

r(v))F_v =⇒ ψ_rf_st =f_ut

Change notation: standard tableaux are determined by their contents so let’s replace twith its content sequence

c(t) = c₁(t),c₂(t), . . . ,c_n(t)

Let I ={z ·1_k ∈Z| −n≤ z ≤n}. Then c(t)∈Iⁿ. Generalising the definition of Ft, forc∈ Iⁿ define

Fc =

n

Y

r=1

Y

d∈Iⁿ cr6=dr

Lr−dr

cr−dr

Acting on {f_st},F_c 6=0 if and only ifc=c(t), for somet∈Std(P_n^Λ)

Semisimple KLR algebras of type A

Theorem

The algebrakSn is generated by {Fc|c∈Iⁿ} ∪ {ψ1, . . . , ψ_n−1}subject to the relations

F_cF_d =δ_cdF_c, P

c∈IⁿF_c =1, ψ_rF_c= F_s_r_cψ_r ψ_r²Fc =δ_c_r_6=c_r₊₁Fc, ψ_rψ_t =ψ_tψ_r if |r −t|>1

(ψ_r+1ψ_rψ_r+1−ψ_rψ_r+1ψ_r)F_c =







Fc, if c_r+2=c_r −→c_r+1,

−F_c, if c_r+2=c_r ←−c_r+1, 0, otherwise

Proof Using the seminormal form it is straightforward to check that these relations hold in kSn. Given this it is easy to deduce thatkSn is

isomorphic to the abstract algebra with the presentation above.

Remark In the semisimple case, Rn^Λ is concentrated in degree zero, so we are not seeing an interesting grading on kS_n yet.

Remark This argument works, essentially without change for all of the algebras Hn^Λ. We need only define thecontent of a standard `-tableau to bec_m(t) = [κ_l +c−r]_ξ if t(l,r,c) =m, for1≤m≤n

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Residue sequences

Now suppose that kis a field of characteristic p, diving n. Then the primitive idempotents F_t∈QS_n cannot, in general, be reduced mod p to give elements of kS_n because of the denominators in their definition.

Similarly, the Jucys-Murphy elements L_k no longer act as scalars but as upper triangular matrices.

Let I =Z/pZ. The residue sequence of a standard tableaut is the sequence i^t= (i₁^t, . . . ,i_n^t)∈ Iⁿ, wherei_k =c_k(t) +pZ. Like contents, residues increase along rows and decrease down columns, mod p Example If λ= (4,3,3,2)and p=3 then the residues in[λ] are:

0 1 2 0 2 0 1 1 2 0 0 1

Given i∈Iⁿ let Std(i) ={t standard|i^t=i}. Frequently, Std(i) =∅

Lifting idempotents

For i∈Iⁿ let F_i = X

t∈Std(i)

F_t∈QS_n

Proposition

Suppose i∈Iⁿ. Then F_i ∈Z(p)Sn

Proof Let F_t⁰ =

n

Y

r=1

Y

s∈StdP_n^Λ i_r^s6=i_r^t

Lr−cr(s)

cr(t)−cr(s) ∈ OS_n

=⇒ F_t⁰ =F_t⁰ X

s∈Std(P_n^Λ)

F_s= X

s∈Std(i)

a_stF_s, for some a_st ∈Z(p)

In particular,a_tt = 1andF_iF_t⁰ =F_t⁰. Therefore, sinceF_sF_u= δ_suF_s, Y

t

(F_i−F_t⁰) =Y

t

X

s6=t

(1−a_st)F_s

= 0

=⇒ F_i = Y

t∈Std(i)

(F_i−F_t⁰) − X

∅6=S⊆Std(i)

(−1)^|S|Y

s∈S

F_s⁰ ∈Z(p)S_n

Andrew Mathas—Cyclotomic quiver Hecke algebras II 2 14 / 22

The KLR generators in Z

(p)

S

_n

The idempotents F_i take care of the “semisimple” elements in Ln^Λ

For each i ∈I fix ˆi ∈Z such thati = ˆi +pZ. The nilpotent elements in Ln^Λ are,y_r =P

i∈Iⁿ

P

t∈Std(i) L_r −ˆi_r

F_t, Now considerψ_r:

ψ_r = X

v∈Std(P_n^Λ)

s_r− _ρ¹

r(v)

₁

βr(v)F_v

Take β_r(v) = (1+ρ_r(v))/ρ_r(v). Then ψ_r becomes

ψ_r = X

v∈Std(P_n^Λ)

(s_rρ_r(v)−1)_1+ρ¹

r(v)F_v

= X

v∈Std(P_n^Λ)

s_r(L_r+1−L_r )−1 ₁

1+L_r+1−LrF_v

= L_rs_r−s_rL_r) X

v∈Std(P_n^Λ) 1 1+Lr+1−LrF_v

The right-hand side makes sense as an element of Z(p)S_n provided that 1+i_r+1^v −i_r^v∈/pZ. If i_r^v= i_r+1^v then (L_rs_r−s_rL_r)F_i =pZ(p)S_n.

The graded isomorphism theorem

Theorem (Brundan-Kleshchev, Hu-M.)

Suppose that k= Z(p). For1≤r <n and i∈Iⁿ define y_r =P

i∈Iⁿ

P

t∈Std(i) L_r −ˆi_r

F_t and

ψ_rF_i =











(s_r+1) ¹

Lr+1−LrF_i, if i_r =i_r+1, (L_rs_r−s_rL_r)F_i, if i_r =i_r+1+1, (L_rs_r−s_rL_r)_L ¹

r+1−LrF_i, otherwise

Theny_r, ψ_r,F_i ∈kS_n. These elements generate kS_n and they induce an isomorphism kSn ∼=Rn^Λ(k).

To prove this it is enough the relations on the seminormal basis

ofQS_n, which is completely straightforward. To complete the proof that kS_n ∼=Rn^Λ you can use a dimension count, which comes from the categorification of the Fock space

This shows thatRn^Λ is an “idempotent completion” of kS_n: once the idempotents F_i belong to Hn^Λ(k)then algebra becomes isomorphic toRn^Λ(k)

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A graded cellular basis of k S

_n

The KLR generators of Rn^Λ, which induce its grading, are ψ1, . . . , ψ_n−1, y1, . . . ,yn, 1_i, for i ∈Iⁿ Theorem (Hu-M.)

Suppose that k is a field, Then kS_n is a graded cellular algebra with graded cellular basis {ψst|s,t∈Std(λ) andλ∈ P_n^Λ}.

Example Take p= 3andλ= (7,5,3). The initial λ-tableaut^λ has the numbers 1,2, . . . ,n entered in order along the rows ofλ:

t^λ =

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

and

0 1 2 0 1 2 0 2 0 1 2 0 1 2 0 Then ψ_tλt^λ =1_iλy^λ, where

i^t^λ = (0,1,2,0,1,2,0,2,0,1,2,0,1,2,0) and y_λ= y₃y₆y₁₀y₁₅

In general, ψ_st =ψ_d(s)−11_iλy^λψ_d(t), wheres=t^λd(s)and t= t^λd(t).

Cellular algebras

Let A be an unital k-algebra, where kis a commutative ring with one Definition (Graham and Lehrer, 1996)

A cellular basis for A is a triple (C,P,S), whereP is a poset with order >, S(λ)is a finite set for λ∈ P and

C: a

λ∈P

S(λ)×S(λ)−→A; (s,t)7→c_st^λ is an injective map such that

1 {c_st^λ |λ∈ P,s,t∈S(λ)} is ak-basis of A

2 If a∈A then ac_st^λ ≡P

u∈Sr_su(a)c_ut^λ (mod A^>λ), where r_su(a) does not depend on t andA^>λ is the subspace ofA spanned by

{cuv^µ |µ > λ and u,v ∈S(µ)}

3 The map ∗:A−→A;c_st^λ 7→c_ts^λ is an anti-isomorphism A cellular algebra is an algebra that has a cellular basis

If A is a graded algebra then a cellular basis(C,P,S)of A is agraded cellular basisif, in addition, there exists a degree function

deg :`

λ∈PS(λ)−→Z;t7→degtsuch that degc_st^λ = degs+ degt

Cellular algebra examples

1 Let A=Matn(k)be the algebra of n×n matrices. Take P ={#}, S(#) ={1,2, . . . ,n} and c_ij^# =e_ij,

where eij is the elementary matrix with 1in row i and columnj and0 elsewhere. ThenA is cellular because

e_ije_kl =δ_jke_il

2 Let {f_st|(s,t)∈Std²(P_n^Λ)}be a seminormal basis of kS_n. This is a cellular basis because f_stf_uv =δ_tvγ_tf_sv

The basis ψ_st is cellular essentially because ψ_st =f_st+ higher terms

Graded Specht modules – cellular algebras

One of the main properties of a cellular basis is that hψ_sv= X

a∈Std(λ)

r_sa(h)ψ_av (mod higher shapes)

The gradedSpecht moduleS^λ has basis {ψ_t|t∈ Std(λ)} andRn^Λ-action hψ_s= X

a∈Std(λ)

r_sa(h)ψ_a

Importantly, S^λ has a natural homogeneous bilinear form h, i Consider: ψ_stψ_uv=hψ_t, ψ_uiψ_sv

=⇒ radS^λ ={x ∈S^λ| hx,yi= 0 for all y ∈S^λ}is a graded submodule of S^λ as hxh,yi=hx,yh^∗i is homogeneous Define D^µ= S^µ/radS^µ, a graded quotient of S^µ

Theorem (Brundan-Kleshchev, Hu-M.)

Over a field,{D^µhki |µ∈ K^Λ_n andk ∈Z}is a complete set of pairwise non-isomorphic irreducible kS_n-modules. Moreover, (D^µ)^~∼= D^µ.

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Cyclotomic quiver Hecke algebras II