Affine Cellularity of Khovanov-Lauda-Rouquier algebras in type A
aa r X i v : . [ m a t h . R T ] O c t AFFINE CELLULARITY OFKHOVANOV-LAUDA-ROUQUIER ALGEBRAS IN TYPE A
ALEXANDER S. KLESHCHEV, JOSEPH W. LOUBERT,AND VANESSA MIEMIETZAbstract.
We prove that the Khovanov-Lauda-Rouquier algebras R α of type A ∞ are (graded) affine cellular in the sense of Koenig and Xi.In fact, we establish a stronger property, namely that the affine cellideals in R α are generated by idempotents. This in particular impliesthe (known) result that the global dimension of R α is finite, and yieldsa theory of standard and reduced standard modules for R α . Introduction
The goal of this paper is to establish (graded) affine cellularity for theKhovanov-Lauda-Rouquier algebras R α of type A ∞ in the sense of Koenigand Xi [ ]. In fact, we construct a chain of affine cell ideals in R α whichare generated by idempotents. This stronger property is analogous to quasi-heredity for finite dimensional algebras, and by a general result of Koenigand Xi [ , Theorem 4.4], it also implies finite global dimension of R α . Thuswe obtain a new proof of a recent result of Kato [ ] and McNamara [ ] intype A over an arbitrary field. As another application, we automatically geta theory of standard and reduced standard modules, cf. [ ].The (finite dimensional) cyclotomic quotients of R α have been shown tobe graded cellular by Hu and Mathas [ ]. Their proof uses the isomorphismtheorem from [ ], the ungraded cellular structure constructed in [ ], and theseminormal forms of cyclotomic Hecke algebras. The affine cellular structurethat we construct here is combinatorially less intricate and does not appealto seminormal forms.Our affine cellular basis is built from scratch, using only the defining rela-tions, some weight theory from [ ], and a dimension formula [ , Theorem4.20]. It is not clear whether it can be deduced from the basis in [ ] bya limiting procedure. At any rate, our philosophy is that one should firstconstruct affine cellular structures and then ‘project’ them to the quotients.This seems to be the only approach available for Lie types other than A .We now give a definition of (graded) affine cellular algebra from [ ,Definition 2.1]. For this introduction, we fix a noetherian domain k (lateron it will be sufficient to work with k = Z ). By definition, an affine algebra is a quotient of a polynomial algebra k [ x , . . . , x n ] for some n .Throughout the paper, unless otherwise stated, we assume that all al-gebras are ( Z )-graded, all ideals, subspaces, etc. are homogeneous, and allhomomorphisms are homogeneous degree zero homomorphisms with respectto the given gradings. Research supported in part by the NSF grant no. DMS-1161094, the Humboldt Foun-dation, and the ERC grant PERG07-GA-2010-268109. The paper has been completed atthe University of Stuttgart. The authors thank Steffen Koenig for hospitality.
Let A be a (graded) unital k -algebra with a k -anti-involution τ . A (two-sided) ideal J in A is called an affine cell ideal if the following conditionsare satisfied:(i) τ ( J ) = J ;(ii) there exists an affine k -algebra B with a k -involution σ and a free k -module V of finite rank such that ∆ := V ⊗ k B has an A - B -bimodule structure, with the right B -module structure induced bythe regular right B -module structure on B ;(iii) let ∆ ′ := B ⊗ k V be the B - A -bimodule with left B -module structureinduced by the regular left B -module structure on B and right A -module structure defined by( b ⊗ v ) a = s( τ ( a )( v ⊗ b )) , (1.1)where s : V ⊗ k B → B ⊗ k V, v ⊗ b → b ⊗ v ; then there is an A - A -bimodule isomorphism α : J → ∆ ⊗ B ∆ ′ , such that the followingdiagram commutes: J α / / τ (cid:15) (cid:15) ∆ ⊗ B ∆ ′ v ⊗ b ⊗ b ′ ⊗ w w ⊗ σ ( b ′ ) ⊗ σ ( b ) ⊗ v (cid:15) (cid:15) J α / / ∆ ⊗ B ∆ ′ . The algebra A is called graded affine cellular if there is a k -module decom-position A = J ′ ⊕ J ′ ⊕ · · · ⊕ J ′ n with τ ( J ′ l ) = J ′ l for 1 ≤ l ≤ n , such that,setting J m := L ml =1 J ′ l , we obtain an ideal filtration0 = J ⊂ J ⊂ J ⊂ · · · ⊂ J n = A so that each J m /J m − is an affine cell ideal of A/J m − .To describe our main results we introduce some notation referring thereader to the main body of the paper for details. Let Q + be the non-negativeroot lattice corresponding to the root system of type A ∞ , α ∈ Q + of height d , and R α be the corresponding KLR algebra with standard generators e ( i ) , ψ , . . . , ψ d − , y , . . . , y d . We denote by Π( α ) be the set of root partitionsof α , see Section 2.3. To any π ∈ Π( α ) we associate the Young subgroup S π ≤ S d and denote by S π the set of the shortest left coset representativesfor S π in S d . We define the polynomial subalgebras Λ π ⊆ R α – these areisomorphic to tensor products of algebras of symmetric polynomials, see(5.2). We also define the monomials y π ∈ R α and idempotents e π ∈ R α , seeSection 5.1. Then we set I ′ π := k - span { ψ w y π Λ π e π ψ τv | w, v ∈ S π } ,I π := P σ ≥ π I ′ σ , and I >π = P σ>π I ′ σ . Our main results are now as follows: Main Theorem.
The algebra R α is graded affine cellular with cell chaingiven by the ideals { I π | π ∈ Π( α ) } . Moreover, setting ¯ R α := R α /I >π for afixed π ∈ Π( α ) , we have: (i) the map Λ π → ¯ e π ¯ R α ¯ e π , b ¯ b ¯ e π is an isomorphism of graded alge-bras; (ii) ¯ R α ¯ e π is a free right ¯ e π ¯ R α ¯ e π -module with basis { ¯ ψ w ¯ y π ¯ e π | w ∈ S π } ; (iii) ¯ e π ¯ R α is a free left ¯ e π ¯ R α ¯ e π -module with basis { ¯ e π ¯ ψ τv | v ∈ S π } ; FFINE CELLULARITY OF KLR ALGEBRAS IN TYPE A 3 (iv) multiplication provides an isomorphism ¯ R α ¯ e π ⊗ ¯ e π ¯ R α ¯ e π ¯ e π ¯ R α ∼ −→ ¯ R α ¯ e π ¯ R α ;(v) ¯ R α ¯ e π ¯ R α = I π /I >π . Main Theorem(v) shows that each affine cell ideal I π /I >π in A/I >π isgenerated by an idempotent. This, together with the fact that each algebraΛ π is a polynomial algebra, is enough to invoke [ , Theorem 4.4] to get Corollary.
If the ground ring k has finite global dimension, then the algebra R α has finite global dimension. This seems to be a slight generalization of [ ] and [ ] (in type A only)in two ways: Kato works over fields of characteristic zero, and McNamaraseems to work over arbitrary fields; moreover, [ ] and [ ] deal with cate-gories of graded modules only, while our corollary holds for the algebra R α even as an ungraded algebra.In the following conjectures we use the term graded affine quasi-hereditary to denote the graded affine cellular algebras with the affine cell ideals satis-fying the additional nice properties described in Main Theorem. Conjecture. (i)
All Khovanov-Lauda-Rouquier algebras are graded affinecellular. (ii)
All cyclotomic Khovanov-Lauda-Rouquier algebras are graded cellular. (iii)
Let us fix a Lie type Γ . Then the Khovanov-Lauda-Rouquier algebras R α (Γ) are graded affine quasi-hereditary for all α ∈ Q + if and only if Γ isof finite type. In [ ], we prove this conjecture for finite simply laced Lie types Γ.The organization of the paper is as follows. Section 2 is preliminary.Section 3 establishes a graded dimension formula for R α , which is later usedto show that the elements of our affine cellular basis are actually linearlyindependent. Section 4 deals with the special case of the affine nilHeckealgebra. This case will be fed into the proof of the general case. Finally, inSection 5, we prove the main results.2. Preliminaries
Lie theoretic notation.
Let Γ be the Dynkin quiver of type A ∞ withthe set of vertices I = Z and the corresponding Cartan matrix a i,j := i = j, | i − j | > , − i = j ± i, j ∈ I . We have a set of simple roots { α i | i ∈ I } and the positive partof the root lattice Q + := L i ∈ I Z ≥ α i . The set of positive roots is given by { α ( m, n ) := α m + α m +1 + · · · + α n | m, n ∈ I, m ≤ n } . For α = P i ∈ I c i α i ∈ Q + , we denote by | α | := P i ∈ I c i the height of α . Wefurthermore have a set of fundamental weights { ω i | i ∈ I } and the set ofdominant weights P + := L i ∈ I Z ≥ ω i .The symmetric group S d with basic transpositions s , . . . , s d − acts onthe set I d by place permutation. The orbits are the sets h I i α := { i = ( i , . . . , i d ) ∈ I d | α i + · · · + α i d = α } ALEXANDER KLESHCHEV, JOSEPH LOUBERT, AND VANESSA MIEMIETZ for each α ∈ Q + with | α | = d . We let ≥ denote the lexicographic order on h I i α determined by the natural order on I = Z .To a positive root β = α ( m, n ), we associate the word i β := ( m, m + 1 , . . . , n ) ∈ h I i β . We denote the set of positive roots by Φ + . Define a total order on Φ + by β ≤ γ if and only if i β ≤ i γ ( β, γ ∈ Φ + ) . (2.2)2.2. KLR Algebras.
For α ∈ Q + of height d and the commutative unitalground ring k , let R α = R α ( k ) denote the associative, unital k -algebra ongenerators { e ( i ) | i ∈ h I i α } ∪ { y , . . . , y d } ∪ { ψ , . . . , ψ d − } subject to thefollowing relations e ( i ) e ( j ) = δ i , j e ( i ); X i ∈h I i α e ( i ) = 1; y r e ( i ) = e ( i ) y r ; ψ r e ( i ) = e ( s r · i ) ψ r ; y r y s = y s y r ; ψ r y s = y s ψ r if s = r, r + 1; ψ r ψ s = ψ s ψ r if | r − s | > ψ r y r +1 e ( i ) = ( y r ψ r + δ i r ,i r +1 ) e ( i ); y r +1 ψ r e ( i ) = ( ψ r y r + δ i r ,i r +1 ) e ( i ); ψ r e ( i ) = i r = i r +1 ,e ( i ) if | i r − i r +1 | > , ( y r +1 − y r ) e ( i ) if i r = i r +1 + 1 , ( y r − y r +1 ) e ( i ) if i r = i r +1 − ψ r ψ r +1 ψ r e ( i ) = ( ψ r +1 ψ r ψ r +1 + 1) e ( i ) if i r +2 = i r = i r +1 + 1 , ( ψ r +1 ψ r ψ r +1 − e ( i ) if i r +2 = i r = i r +1 − ,ψ r +1 ψ r ψ r +1 e ( i ) otherwise . There is a unique Z -grading on R α such that all e ( i ) are of degree 0, all y r are of degree 2, and deg( ψ r e ( i )) = − a i r ,i r +1 (see 2.1).Fixing a reduced decomposition w = s r . . . s r m for each w ∈ S d , we definethe elements ψ w := ψ r . . . ψ r m ∈ R α for all w ∈ S d . Theorem 2.3. [ , Theorem 2.5] , [ , Theorem 3.7] A k -basis of R α isgiven by { ψ w y m . . . y m d d e ( i ) | w ∈ S d , m , . . . , m d ∈ Z ≥ , i ∈ h I i α } . The commutative subalgebra of R α generated by y , . . . , y d is thus iso-morphic to the polynomial algebra k [ y , . . . , y d ] and will be denoted by P d .In view of Theorem 2.3, we have R α ( k ) ≃ R α ( Z ) ⊗ Z k , so in what followswe can work with k = Z . When we need to deal with representation theoryof R α we often assume that k is a field.We will also use the diagrammatic notation introduced in [ ] to representelements of R α . Given i = ( i , . . . , i d ) ∈ h I i α , we write e ( i ) = i i i d , ψ r e ( i ) = i i r − i r i r +1 i d , y s e ( i ) = i i s − i s i s +1 i d where 1 ≤ r < d and 1 ≤ s ≤ d .We say that w ∈ S d possesses a left-right symmetric reduced decomposi-tion , if we can write w = s r . . . s r m = s r m . . . s r and the reduced expression FFINE CELLULARITY OF KLR ALGEBRAS IN TYPE A 5 s r m . . . s r can be obtained from s r . . . s r m by using only commuting Coxeterrelations. Lemma 2.4.
The element w ∈ S d possesses a left-right symmetric reduceddecomposition.Proof. Since reversing the order of the basic transpositions in a reduceddecomposition amounts to reflecting the representing braid diagram acrossthe horizontal, it is easy to see thatgives rise to a left-right symmetric reduced decomposition as desired. (cid:3)
Root partitions.
Let α ∈ Q + . A root partition of α is a way towrite α as an ordered sum of positive roots α = p β + · · · + p N β N so that β > · · · > β N and p , . . . , p N >
0. We denote such a root partition π asfollows: π = β p . . . β p N N . (2.5)The set of all root partitions of α is denoted Π( α ).To a root partition π as in (2.5) we associate the word i π := i β . . . i β . . . i β N . . . i β N ∈ h I i α as the concatenation of the i β k where each i β k occurs p k times. Define thetotal order on Π( α ) via π ≥ σ if and only if i π ≥ i σ for π, σ ∈ Π( α ).To a root partition π as in (2.5) we also associate a parabolic subgroup S π ≤ S d : S π ∼ = S × p | β | × · · · × S × p N | β N | and the set S π of the minimal length left coset representatives of S π in S d .The orbits of S π on { , . . . , d } will be referred to as π -blocks . The first p ofthe π -blocks are of size | β | , and will be referred to as the π -blocks of weight β , the next p of the π -blocks are of size | β | , and will be referred to as the π -blocks of weight β , etc.2.4. Representation theory.
Set A := Z [ q, q − ]. For a graded vectorspace V = ⊕ n ∈ Z V n we set dim q V := P n ∈ Z (dim V n ) q n . If f = P m ∈ Z a m q m ∈ A , we denote deg n ( f ) := a n q n . We denote by V h m i the graded vector spacewith degrees shifted up by m so that V h m i n = V n − m .We will use an operation of induction on the KLR-algebras defined in [ ].Given an R α -module M and an R β -module N , we thus have an inducedmodule Ind α,β M ⊠ N over R α + β , which will also be denoted M ◦ N .For a positive root β , there is a unique one-dimensional R β -module L ( β )with e ( i β ) L ( β ) = 0 and all other generators acting as zero. For a rootpartition π ∈ Π( α ) as in (2.5) we set sh ( π ) := P Nk =1 p k ( p k − / , 7.1], define the (reduced) standard module ¯∆( π ) := L ( β ) ◦ p ◦ · · · ◦ L ( β N ) ◦ p N h sh ( π ) i . Let k be a field. By [ , Theorem 7.2], ¯∆( π ) has a unique irreduciblequotient, denoted by L ( π ), and { L ( π ) | π ∈ Π( α ) } is a complete systemof (graded) irreducible R α -modules up to isomorphism. Furthermore, i π is ALEXANDER KLESHCHEV, JOSEPH LOUBERT, AND VANESSA MIEMIETZ lexicographically the largest among the words i ∈ h I i α such that e ( i ) L ( π ) =0.2.5. Poincar´e polynomials.
We will make use of the following well-knowncomputation of the Poincar´e polynomial, see e.g. [ , Theorem 3.15]: Lemma 2.6.
We have X w ∈ S a t ℓ ( w ) = a Y r =1 t r − t − t a ( a − / a Y r =1 − t − r − t − . A dimension formula
In this section we establish a graded dimension formula for R α . Thisformula can be thought of as a combinatorial shadow of the affine cellularstructure on R α to be constructed later. We point out that there is a similardimension formula for any finite type KLR algebra [ ]. The proof we givehere works for type A only, but it might be of independent interest since itexploits a ‘limiting procedure’ and the dimension formula from [ , Theorem4.20]. By Theorem 2.3, the graded dimension of R α ( k ) does not depend on k , so in this section we fix k a field.We start by the following observation: Lemma 3.1.
Let β = α i + · · · + α j and γ = α i + · · · + α k for j > k . Then L ( β ) ◦ L ( γ ) ≃ L ( γ ) ◦ L ( β ) h i is irreducible.Proof. It is easy to see that i β i γ is the only dominant weight in L ( β ) ◦ L ( γ ),and it appears with multiplicity one. The result easily follows, cf. [ ]. (cid:3) Cyclotomic KLR-algebras.
For the rest of this section we fix α ∈ Q + of height d . Let Ω = X i ∈ I b i ω i (3.2)be a dominant weight of level l := P i ∈ I b i , and consider the correspondingcyclotomic quotient R Ω α . We will use the notation and results of [ , ].In particular, by P Ω α is the set of all l -multipartitions of weight α , cf. [ ,(3.15)], for λ ∈ P Ω α , we denote by T ( λ ) the set of standard λ -tableaux,deg( S ) denotes the degree of the standard tableau S ∈ T ( λ ), cf. [ , Section4.11] [ , Section 3.2], and S λ denotes the Specht module corresponding to λ , cf. [ , Section 4.2]. The definition of deg( S ) depends on the choice of amulticharge κ = ( k , . . . , k l ) such that ω k + · · · + ω k l = Ω, cf. [ , Section 3].We always make the choice for which k ≥ · · · ≥ k l . By [ , Theorem 4.20],we have dim q R Ω α = X λ ∈ P Ω α (cid:16) X S ∈ T ( λ ) q deg S (cid:17) . By [ , Corollary 3.14], we can rewrite this as follows:dim q R Ω α = X λ ∈ P Ω α (cid:16) X S ∈ T ( λ ) q deg( ψ w S e ( i λ ))+deg( T λ ) (cid:17) , (3.3)where T λ is the leading λ -tableau, i λ ∈ h I i α is the corresponding residuesequence, and w S is defined by w S T λ = S , cf. [ , Section 3.2]. FFINE CELLULARITY OF KLR ALGEBRAS IN TYPE A 7
The symmetric group S l acts on l -multipartitions by permuting theircomponents, so that w · λ = ( λ ( w − (1)) , . . . , λ ( w − ( l )) ). The parabolic sub-group S (Ω) = × i ∈ I S b i ≤ S l then acts on P Ω α . Let λ = ( λ (1) , . . . , λ ( l ) ) ∈ P Ω α be such that each λ ( m ) is a non-trivial one-row partition. Let β m = P b ∈ λ ( m ) α res( b ) where the summation is over all boxes b of λ ( m ) and res( b ) ∈ I denotes the residue of b . There exists an element w ∈ S l such that β w − (1) ≥ · · · ≥ β w − ( l ) . Let ℓ λ be the length of the shortest such element,and define π ( λ ) := β w − (1) . . . β w − ( l ) ∈ Π( α ) . By inflation we consider all R Ω α -modules as R α -modules. We want toconnect standard modules to some special Specht modules. Proposition 3.4.
Let λ = ( λ (1) , . . . , λ ( l ) ) ∈ P Ω α be such that each λ ( m ) is anon-trivial one-row partition. Denote λ op := w Ω0 · λ , where w Ω0 is the longestelement in S (Ω) . Then S λ op ≃ ¯∆( π ( λ )) h ℓ λ i Proof.
This follows from Lemma 3.1 and [ , Theorem 8.2] (cid:3) For a root partition π as in (2.5) and a positive integer p , denote c π := q sh ( π ) X w ∈ S π q deg ψ w e ( i π ) ,l p := p Y m =1 − q m ,l π := N Y k =1 l p k . Note that c π is the dimension of the reduced standard module ¯∆( π ) and l p is the dimension of the algebra Λ p of symmetric polynomials in p variablesof degree 2.3.2. The formula.
Our dimension formula is now as follows:
Proposition 3.5.
We have dim q R α = X π ∈ Π( α ) l π c π . Proof.
Let us fix n ∈ Z . It suffices to prove thatdeg n (dim q R α ) = deg n ( X π ∈ Π( α ) l π c π ) . Note that we can choose b i ≫ i in the support of α , such thatdeg n (dim q R α ) = deg n (dim q R Ω α ). Let us make this choice and prove thatdeg n (dim q R Ω α ) = deg n ( P π ∈ Π( α ) l π c π ). Claim 1.
Let R be the set of all multipartitions λ = ( λ (1) , . . . , λ ( l ) ) ∈ P Ω α such that each λ ( a ) is either empty or one row. Thendeg n (dim q R Ω α ) = deg n (cid:16) X λ ∈ R (cid:16) X S ∈ T ( λ ) q deg( ψ w S e ( i λ ))+deg( T λ ) (cid:17) (cid:17) . Proof of Claim 1.
Note that deg( ψ w S e ( i λ )) ≥ − d !. So in view of (3.3), itsuffices to prove that deg( T λ ) ≫ n unless S ∈ R . Let λ ∈ P Ω α . If λ ( m ) = ∅ , ALEXANDER KLESHCHEV, JOSEPH LOUBERT, AND VANESSA MIEMIETZ then k m is in the support of α . If k m − α , then λ ( m ) can only have one row. If k m − α and λ has atleast two rows, then deg( T λ ) ≥ b k m − ≫
0. Claim 1 is proved.Next, let λ be a multipartition in R . Set n i ( λ ) := ♯ { m | λ ( m ) = ∅ and k m = i } . Claim 2.
Let Θ ⊆ R be the subset of all multipartitions λ ∈ R such thatwhenever λ ( m ) = ∅ , then λ ( a ) = ∅ for all a > m with k a = k m . Thendeg n (dim q R Ω α ) = deg n (cid:16) X λ ∈ Θ (cid:16) X S ∈ T ( λ ) q deg( S ) (cid:17) Y i ∈ I l n i ( λ ) (cid:17) . Proof of Claim 2.
Let λ ∈ R and let λ + ∈ Θ be the multipartition obtainedby shifting the non-empty components λ ( m ) of λ corresponding to m withthe same k m (without changing the order of the non-empty components).To be more precise, for each i with b i = 0, each nonempty component λ ( m ) with k m = i gets moved to a larger position m + γ i ( m ). Note that γ i ( m ) ≤ γ i ( m ′ ) whenever m > m ′ with k m = k m ′ = i and λ ( m ) , λ ( m ′ ) arenon-empty. This defines a multipartition γ = ( γ i ) i ∈ I , where each partition γ i has at most n i ( λ ) parts. For S ∈ T ( λ ) let S + ∈ T ( λ + ) be the correspondingtableau obtained from S by the same shift which takes λ to λ + . Note thatdeg( S + ) = deg( S ) P i ∈ I | γ i | . Let p n ( t ) be the generating function for thepartitions with at most n parts. Note that l n i ( λ ) = p n i ( λ ) ( q ). Now Claim 2follows.We now finish the proof of the proposition. DenoteΘ( π ) := { λ ∈ Θ | π ( λ ) = π } . For λ ∈ Θ( π ), observe that the group S π is naturally a parabolic subgroupof G := × i ∈ I S n i ( λ ) , giving an equality for Poincar´e polynomials P G ( t ) = P S π ( t ) X λ ∈ Θ( π ) t ℓ λ , where ℓ λ is defined before Proposition 3.4. This implies l π = X λ ∈ Θ( π ) q ℓ λ Y i ∈ I l n i ( λ ) . Now, using Proposition 3.4, we have X λ ∈ Θ (cid:16) X S ∈ T ( λ ) q deg( S ) (cid:17) Y i ∈ I l n i ( λ ) = X λ ∈ Θ (cid:16) dim q S λ op (cid:17) Y i ∈ I l n i ( λ ) = X λ ∈ Θ (cid:16) dim q ¯∆( π ( λ )) h ℓ λ i (cid:17) Y i ∈ I l n i ( λ ) = X λ ∈ Θ q ℓ λ c π Y i ∈ I l n i ( λ ) = X π ∈ Π( α ) X λ ∈ Θ( π ) q ℓ λ c π Y i ∈ I l n i ( λ ) = X π ∈ Π( α ) c π l π , as desired. (cid:3) FFINE CELLULARITY OF KLR ALGEBRAS IN TYPE A 9 Affine nilHecke algebra
In this section we will review mostly well-known facts about the nilHeckealgebra, and obtain a special case of our main result for this algebra. Thisspecial case will be needed in the proofs of the general case.4.1.
Definition and basic properties.
We denote a th nilHecke algebraby H a . That is, H a is the associative, unital ( Z -)algebra generated by { y , . . . , y a , ψ , . . . , ψ a − } subject to the relations ψ r = 0 (4.1) ψ r ψ s = ψ s ψ r if | r − s | > ψ r ψ r +1 ψ r = ψ r +1 ψ r ψ r +1 (4.3) ψ r y s = y s ψ r if s = r, r + 1 (4.4) ψ r y r +1 = y r ψ r + 1 (4.5) y r +1 ψ r = ψ r y r + 1 . (4.6)For w ∈ S a , pick any reduced decomposition w = s i . . . s i k . We define ψ w = ψ i . . . ψ i k . In view of the relations above, ψ w does not depend on thechoice of reduced decomposition. We define deg( y r ) = 2 and deg( ψ r ) = − H a into a graded algebra.There is an involutive homogeneous degree zero anti-automorphism τ of H a fixing the standard generators of H a . We write h τ instead of τ ( h ) for h ∈ H a . Given a (graded) left H a -module M , we write M τ for the (graded) right H a -module given by twisting with τ . The following result gives standardbases of H a : Theorem 4.7.
We have (i) { ψ w y m . . . y m a a | w ∈ S a , m , . . . , m a ≥ } is a Z -basis of H a . (ii) { y m . . . y m a a ψ w | w ∈ S a , m , . . . , m a ≥ } is a Z -basis of H a .In particular, dim q ( H a ) = 1(1 − q ) a X w ∈ S a q deg( ψ w ) . In view of the theorem we can consider the polynomial algebra P a := Z [ y , . . . , y a ]as a subalgebra of H a . Moreover, letΛ a := Z [ y , . . . , y a ] S a be the algebra of symmetric functions. The following is well-known, seee.g. [ ]. Theorem 4.8.
The center of H a is given by Z ( H a ) = Λ a . The idempotent e a . It is well-known that H a can be realized as thesubalgebra of the endomorphism algebra End Z ( P a ) generated by (multipli-cation by) each y r , and the divided difference operators ψ r ( f ) = f − s r fy r +1 − y r , (4.9)where ( s r ( f ))( y , . . . , y a ) = f ( y , . . . , y r +1 , y r , . . . , y a ). In light of this de-scription there is an H a -module structure on P a ; we shall refer to this modulealso as P a . Let δ a = y y . . . y a − a , and define w ∈ S a to be the longest element. It is noticed in [ , Section2.2] that e a := ψ w δ a is an idempotent. Then ψ w δ a ψ w δ a = ψ w δ a implies e a ψ w = ψ w , (4.10)since by Theorem 4.7(i), δ a is not a zero divisor. We will need the followingfacts coming from the theory of Schubert polynomials, see e.g. [ , Section10.4]. Theorem 4.11. P a is a free Λ a -module with basis { ψ w ( δ a ) | w ∈ S a } .Moreover, ψ w ( δ a ) = 1 . The following two theorems are known, but we sketch their proofs for thereader’s convenience.
Theorem 4.12.
The following things are true: (i) H a P a ∼ −→ H a e a , f f e a . (ii) ( P τa ) H a ∼ −→ e a H a , f e a ψ w f . (iii) Λ a ∼ −→ e a H a e a , f f e a .Proof. (i) By Theorem 4.7(ii), H a e a is spanned by elements of the form y m . . . y m a a ψ w e a . But ψ w ψ w = 0 whenever w = 1, so H a e a is in factspanned by elements of the form y m . . . y m a a e a . By (4.10), we have y m . . . y m a a e a ψ w = y m . . . y m a a ψ w . Since such elements are linearly independent, our spanning set above isactually a basis. In particular, the map P a → H a e a , f f e a is an isomorphism of Z -modules. To show that it is H a -equivariant, notethat the action of y r is preserved, and furthermore ψ r f e a = f ψ r e a + ψ r ( f ) e a = ψ r ( f ) e a , where ψ r ( f ) is the action on P a defined in (4.9).(ii) is proved similarly to (i).(iii) e a H a e a is spanned by the elements e a f e a with f ∈ P a . Using (i) weget e a f e a = ψ w δ a f e a = ψ w ( δ a f ) e a , and ψ w ( δ a f ) ∈ Λ a . We thus see that e a H a e a is spanned by be a with b ∈ Λ a .Rewrite again: be a = e a b = ψ w δ a b . Now, by Theorem 4.7(i),Λ a → e a H a e a , b be a is an isomorphism. (cid:3) Theorem 4.13.
Let ι : H a → End Z ( P a ) be the map which comes from theaction of H a on P a . This map yields an isomorphism of algebras ι : H a ∼ −→ End Λ a ( P a ) . FFINE CELLULARITY OF KLR ALGEBRAS IN TYPE A 11
Proof.
Let x = P u f u ψ u ∈ H a be a non-zero element. Let u be a min-imal element in the Bruhat order with f u = 0. Apply x to the element ψ u − w ( δ a ) ∈ P a , see Theorem 4.11. Then x ( ψ u − w ( δ a )) = f u , which showsthat ι is injective.On the other hand, since Λ a = Z ( H a ), it is clear that the image of ι is contained in End Λ a ( P a ). Using Theorem 4.11 again and comparing thegraded dimensions, we see that ι is an isomorphism. (cid:3) Corollary 4.14.
We have H a e a H a = H a .Proof. By using the basis of Theorem 4.11, ordered so that δ a is the firstelement, we can identify End Λ a ( P a ) with the matrix algebra M n ! (Λ a ). Thenunder the isomorphism ι from the theorem τ ( e a ) gets mapped to the matrixunit E , . The result follows. (cid:3) Affine cellular basis of the nilHecke algebra.Lemma 4.15.
We have: (i) H a e a is free as a right e a H a e a -module with basis { ψ w δ a e a | w ∈ S a } ; (ii) e a H a is free as a left e a H a e a -module with basis { e a ψ τv | v ∈ S a } .Proof. By Theorem 4.11, a basis for P a over Λ a is given by all ψ w ( δ a ) for w ∈ S a . Now by Theorem 4.12(i),(iii), H a e a is free as a right e a H a e a -modulewith basis { ψ w ( δ a ) e a | w ∈ S a } . But ψ w ( δ a ) e a = ψ w δ a e a by Theorem 4.12(i)again.For (ii), we use Theorem 4.12(ii),(iii) instead to conclude that the set { e a ψ w ψ w ( δ a ) | w ∈ S a } is a basis of e a H a as a left e a H a e a -module. Noticethat e a ψ w ψ w ( δ a ) = e a ψ w δ a ψ τw = e a ψ τw , and the result follows. (cid:3) The following theorem gives an affine cellular basis of H a . Theorem 4.16.
Let { b x } x ∈ X be any Z -basis of Λ a . The nilHecke algebra H a has a basis given by { ψ w b x δ a e a ψ τv | v, w ∈ S a , x ∈ X } .Proof. By Lemma 4.15, the image H a e a H a of the multiplication map H a e a ⊗ e a H a e a e a H a → H a e a H a is spanned by the set { ψ w b x δ a e a ψ τv | v, w ∈ S a , x ∈ X } . By Corollary 4.14,this set thus spans H a .Next, we compute the degree d of each element of this spanning set, addup the various q d , and see that this is exactly the graded dimension of H a .This shows that this spanning set must be a basis.The degree of ψ w is − ℓ ( w ); the degree of δ a is a ( a − e a is 0. The graded dimension of Λ a is Q ar =1 11 − q r . Let { b x } x ∈ X be ahomogeneous basis of Λ a (for example, the monomial symmetric functions). Then X v,w ∈ S a ,x ∈ X q deg( ψ w )+deg( b x )+deg( δ a )+deg( e a )+deg( ψ v ) = X w ∈ S a q − ℓ ( w ) ! a Y r =1 − q r ! q a ( a − X v ∈ S a q deg( ψ v ) ! = q − a ( a − a Y r =1 − q r − q ! a Y r =1 − q r ! q a ( a − X v ∈ S a q deg( ψ v ) ! = (cid:18) − q ) a (cid:19) X v ∈ S a q deg( ψ v ) ! , which is dim q ( H a ) by Theorem 4.7, and we are done. (cid:3) Affine cellular structure
Throughout this section we work with a fixed element α ∈ Q + of height d .5.1. Basic definitions.
Let α , . . . , α l be elements of Q + with α + · · · + α l = α . Then we have a natural embedding ι α ,...,α l : R α ⊗ · · · ⊗ R α l ֒ → R α of algebras, whose image is the parabolic subalgebra R α ,...,α l .Define the element ψ α ∈ R α to be ψ α := ( ψ d . . . ψ d − ) . . . ( ψ . . . ψ d +1 )( ψ . . . ψ d ) . In other words, ψ α is a ‘permutation of two α -blocks’ and corresponding tothe following element of S d :Now, let p ∈ Z > . We define ψ α,r := ι ( r − α, α, ( p − r − α (1 ⊗ ψ α ⊗ ∈ R pα (1 ≤ r < p ) . In other words, ψ α,r is a ‘permutation of the r th and ( r + 1) st α -blocks’.Moreover, let w ∈ S p with reduced decomposition w = s i . . . s i m . Definean element ψ α,w := ψ α,i . . . ψ α,i m ∈ R pα . Let also y α,s := ι ( s − α,α, ( p − s ) α (1 ⊗ y d ⊗ ∈ R pα (1 ≤ s ≤ p ) . In other words, y α,s is a ‘dot on the last strand of the s th block of size d ’.Further, define δ α,p := y α, y α, . . . y p − α,p ∈ R pα . We have polynomial algebra and the symmetric polynomial algebra P α,p = Z [ y α, , . . . , y α,p ] and Λ α,p = P S p α,p . FFINE CELLULARITY OF KLR ALGEBRAS IN TYPE A 13
Now, let π = β p . . . β p N N ∈ Π( α )be a root partition. For 1 ≤ k ≤ N and x ∈ R p k β k , we put ι k ( x ) := ι p β + ··· + p k − β k − ,p k β k ,p k +1 β k +1 + ··· + p N β N (1 ⊗ x ⊗ ∈ R α . Define for all 1 ≤ k ≤ N , w ∈ S p k , 1 ≤ r < p k and 1 ≤ s ≤ p k , theelements of R α : ψ k,w := ι k ( ψ β k ,w ) , ψ k,r := ι k ( ψ β k ,r ) , y k,s := ι k ( y β k ,s ) . In other words, ψ k,r is the permutation of the r th and ( r + 1) st β k -blocks,and y k,s is a dot on the final strand in the s th β k -block.Finally, define y π = ι ( δ β ,p ) . . . ι N ( δ β N ,p N ) ,ψ π = ι ( ψ β ,w ) . . . ι N ( ψ β N ,w N ) , where w k is the longest element of S p k for k = 1 , . . . , N . We always make achoice of a reduced decompositon of w k which is left-right symmetric in thesense of Lemma 2.4. This will guarantee that ψ τπ = ψ π . (5.1)Recalling the dominant word i π ∈ h I i α , put e π = ψ π y π e ( i π ) . Also, letΛ π = ι p β ,...,p N β N (Λ β ,p ⊗ · · · ⊗ Λ β N ,p N ) ∼ = Λ p ⊗ · · · ⊗ Λ p N . (5.2)5.2. Cells.
Define I ′ π = Z - span { ψ w y π Λ π e π ψ τv | w, v ∈ S π } I π = X σ ≥ π I ′ σ I >π = X σ>π I ′ σ It will turn out that the I π for π ∈ Π( α ) form a chain of cell ideals. Lemma 5.3.
Let w ∈ S π . Then w · i π = i π if and only if w permutes the π -blocks of weight β k for all k = 1 , . . . , N .Proof. This follows from [ , Lemma 5.3(ii)]. (cid:3) Lemma 5.4. ψ π e ( i π ) and e π commute with elements of Λ π .Proof. It suffices to observe that any ψ k,r e ( i π ) commutes with elements ofΛ π , which easily follows from the relations in R α . (cid:3) Lemma 5.5.
We have τ ( I ′ π ) = I ′ π and τ ( I π ) = I π .Proof. It suffices to prove the first equality. Using the definition of e π andLemma 5.4, we get τ ( ψ w y π Λ π e π ψ τv ) = τ ( ψ w y π Λ π ψ π y π e ( i π ) ψ τv )= τ ( ψ w y π ψ π Λ π y π e ( i π ) ψ τv )= ψ v y π e ( i π )Λ π ψ τπ y π ψ τw = ψ v y π Λ π ψ τπ y π e ( i π ) ψ τw . It suffices to note that ψ τπ = ψ π by (5.1). (cid:3) Ideal filtration.
This subsection is devoted to the proof of the follow-ing theorem.
Theorem 5.6. I π is the two-sided ideal P σ ≥ π R α e ( i σ ) R α . We prove the theorem by downward induction on the lexicographic orderon Π( α ). To be more precise, throughout the subsection we assume that wehave proved that I >π = X σ>π R α e ( i σ ) R α (5.7)and from this prove that I π = P σ ≥ π R α e ( i σ ) R α . When π is the maximalroot partition, the inductive assumption is trivially satisfied. Otherwise, I >π = I σ where σ is the immediate successor of π in the lexicographic order. Lemma 5.8. If i > i π , then e ( i ) ∈ I >π .Proof. If π is the maximal root partition, then there is nothing to prove.Let I be any maximal (graded) left ideal containing I >π . Then R α /I ∼ = L ( σ ) for some σ . If σ > π , then e ( i σ ) ∈ I >π by induction, see (5.7), andsince e ( i σ ) L ( σ ) = 0 we would have IL ( σ ) = I ( R α /I ) = 0, which is acontradiction. We conclude that σ ≤ π .Therefore, since i > i π ≥ i σ and since all of the weights appearing in L ( σ ) are less than or equal to i σ , we have e ( i ) L ( σ ) = 0, which implies that e ( i ) ∈ I . We have shown that e ( i ) is contained in every maximal left idealcontaining I >π .Consider the graded left ideal J := I >π + R α (1 − e ( i )). If J is not allof R α , then it is contained in a maximal left ideal I , which by the previousparagraph contains e ( i ). Since 1 − e ( i ) ∈ J ⊆ I , we conclude that I = R α ,which is a contradiction. Therefore J = R α , and we may write 1 = x + r (1 − e ( i )) for some x ∈ I >π and r ∈ R α . Multiplying on the right by e ( i ),we see that e ( i ) = xe ( i ) ∈ I >π .This argument actually proves the lemma over any field, and then it alsofollows for Z by a standard argument. (cid:3) Corollary 5.9. If w ∈ S π \ { } , then ψ w P d e ( i π ) ⊆ I >π . Proof.
Observe that w · i π > i π , whence e ( w · i π ) ∈ I >π by Lemma 5.8. Now,for any f ∈ P d , we have ψ w f e ( i π ) = e ( w · i π ) ψ w f e ( i π ) ∈ I >π . (cid:3) Recall π -blocks defined in the end of Section 5.1. Corollary 5.10. If y r and y s are in the same π -block, then y r e ( i π ) ≡ y s e ( i π ) (mod I >π ) . Proof. If r and r + 1 are in the same π -block, then s r ∈ S π \ { } , and so( y r − y r +1 ) e ( i π ) = ψ r e ( i π ) ∈ I >π by Corollary 5.9. (cid:3) Let us make the choice of reduced decompositions in S d so that whenever w = w π w π for w π ∈ S π and w π ∈ S π , then we have ψ w = ψ w π ψ w π . (5.11)Recall the nilHecke algebra H a from Section 4. FFINE CELLULARITY OF KLR ALGEBRAS IN TYPE A 15
Lemma 5.12.
For each k = 1 , . . . , N there is a ring homomorphism θ k : H p k → ( e ( i π ) R α e ( i π ) + I >π ) /I >π ,y s y k,s e ( i π ) + I >π (1 ≤ s ≤ p k ) ,ψ r ψ k,r e ( i π ) + I >π (1 ≤ r < p k ) . Proof.
We check the relations (4.1)–(4.6). The relations (4.2) and (4.4) areobvious.To check relation (4.1), we write ψ k,r e ( i π ) = P u ∈ S d ψ u f u e ( i π ) with f u ∈ P d . If u / ∈ S π , then ψ u f u e ( i π ) ∈ I >π by (5.11) and Corollary 5.9. On theother hand, suppose that u ∈ S π is such that f u = 0. By Lemma 5.3, u permutes π -blocks of weight β k . Since ψ k,r only contains crossings betweenthe r th and ( r + 1) st π -blocks of weight β k , the same must be true for ψ u .The only possibilities are ψ u = 1 or ψ u = ψ k,r . Since deg( ψ k,r e ( i π )) = − ψ k,r e ( i π ) ∈ I >π .To check relation (4.3), we write( ψ k,r ψ k,r +1 ψ k,r − ψ k,r +1 ψ k,r ψ k,r +1 ) e ( i π ) = X u ∈ S d ψ u f u e ( i π )with f u ∈ P d , and show as above that ψ u that appear with non-zero f u mustbe of the form 1 , ψ k,r , ψ k,r +1 , ψ k,r ψ k,r +1 or ψ k,r +1 ψ k,r . Sincedeg( ψ k,r ψ k,r +1 ψ k,r e ( i π )) = − i β k = ( i, i + 1 , . . . , j ). We calculate ψ k,r y k,r +1 e ( i π ) using Khovanov-Laudadiagram calculus, as explained in Section 2. In doing so, we will ignore thestrands outside of the r th and ( r + 1) st π -blocks of weight β k . We have: i ji j = i ji j + iji j . The first term is y k,r ψ k,r e ( i π ). The second term is iji j + iji j . The first term is in I >π by Corollary 5.9. Continuing in this way, we obtainthe result. (cid:3) Lemma 5.13.
The maps θ , . . . , θ N from Lemma 5.12 have commuting im-ages, and so define a map θ : H p ⊗ · · · ⊗ H p N → ( e ( i π ) R α e ( i π ) + I >π ) /I >π ,h ⊗ · · · ⊗ h N θ ( h ) . . . θ N ( h N ) . Moreover, the image of θ is contained in I π /I >π .Proof. The first statement is obvious. To prove the statement about theimage of θ , recall the idempotent e p k = ψ w k δ p k ∈ H p k . It is clear that θ ( δ p ⊗ · · · ⊗ δ p N ) = y π e ( i π ) + I >π and θ ( e p ⊗ · · · ⊗ e p N ) = e π + I >π . For each k , choose w k , v k ∈ S p k and b k ∈ Λ p k . Then θ ( ψ w b δ p e p ψ τv ⊗ · · · ⊗ ψ w N b N δ p N e p N ψ τv N )=( ψ ,w . . . ψ N,w N )( ι ( b ) . . . ι N ( b N )) y π e π ( ψ τ ,v . . . ψ τN,v N ) + I >π . The right hand side of this equation is in I π /I ′ π , because ψ ,w . . . ψ N,w N isof the form ψ w for w ∈ S π , ψ τ ,v . . . ψ τN,v N is of the form ψ τv for v ∈ S π ,and ι ( b ) . . . ι N ( b N ) ∈ Λ π . By Theorem 4.16, each H p k is spanned by { ψ w Λ p k δ p k e p k ψ τv | v, w ∈ S p k } . We conclude that im( θ ) ⊆ I π /I >π . (cid:3) Corollary 5.14. e ( i π ) ∈ I π .Proof. Note that e ( i π ) + I >π = θ (1 ⊗ · · · ⊗ ∈ I π /I >π by Lemma 5.13. (cid:3) We come now to the main lemma.
Lemma 5.15. If w ∈ S d and v ∈ S π then ψ w P d e π ψ τv ⊆ I π .Proof. We prove this by upward induction on deg( ψ w e ( i π )), using as theinduction base sufficiently negative degree for which ψ w e ( i π ) = 0, and soalso ψ w P d e π ψ τv = { } ⊆ I π .Let f ∈ P d . By Corollary 5.10, f e ( i π ) ≡ ge ( i π ) (mod I >π ) for some g ∈ Pol := Z [ y , , . . . , y ,p , . . . , y N, , . . . , y N,p N ] . Furthermore, we may assume that g = g . . . g N with each g k ∈ Pol k := Z [ y k, , . . . , y k,p k ] , since Pol ∼ = Pol ⊗ · · · ⊗ Pol N .Denote by b g k ∈ H p k the image of g k under the isomorphism Pol k ∼ −→ P p k .Then θ ( b g e p ⊗ · · · ⊗ c g N e p N ) = g . . . g N e π + I >π . (5.16)On the other hand, by Theorem 4.11, there exist d b k,u ∈ Λ p k such that b g k e p k = X u ∈ S pk d b k,u ψ u ( δ p k ) e p k = X u ∈ S pk d b k,u ψ u δ p k e p k = X u ∈ S pk ψ u d b k,u δ p k e p k , where we have used Theorem 4.12(i) for the second equality and Theorem 4.8for the third equality. Therefore, denoting ψ u ,...,u N := ψ ,u . . . ψ N,u N and b u ,...,u N := b ,u . . . b N,u N θ ( b g e p ⊗· · ·⊗ c g N e p N ) = X u ∈ S p ,...,u N ∈ S pN ψ u ,...,u N b u ,...,u N y π e π + I >π . (5.17)We now equate the right hand sides of (5.16) and (5.17) and multiply by ψ w on the left and by ψ τv on the right to obtain ψ w ge π ψ τv ≡ X u ∈ S p ,...,u N ∈ S pN ψ w ψ u ,...,u N b u ,...,u N y π e π ψ τv (mod I >π ) . FFINE CELLULARITY OF KLR ALGEBRAS IN TYPE A 17
We are therefore reduced to proving that each summand on the right handside belongs to I π . Write, using Theorem 2.3, ψ w ψ u ,...,u N e ( i π ) = X x ∈ S d ψ x f x e ( i π )so that ψ w ψ u ,...,u N b u ,...,u N y π e π ψ τv = X x ∈ S d ψ x f x b u ,...,u N y π e π ψ τv . (5.18)If at least one of u , . . . , u N is not 1, then deg( ψ u ,...,u N e ( i π )) <
0, and sodeg( ψ x e ( i π )) ≤ deg( ψ x f x e ( i π )) < deg( ψ w e ( i π )) . Therefore we are done by induction in this case.Now, let u = · · · = u N = 1. By (5.11), we can write ψ w = ψ w ψ w with w ∈ S π and w ∈ S π . If w = 1, then we are done by Corollary 5.9.Otherwise, recalling that b u ,...,u N ∈ Λ π , we have ψ w b u ,...,u N y π e π ψ τv ∈ I ′ π bydefinition. (cid:3) Corollary 5.19. I π is an ideal.Proof. By Lemma 5.5, we have τ ( I π ) = I π . So it is enough to prove that I π is a left ideal. Now, for x ∈ R α and w, v ∈ S π , we have xψ w y π Λ π e π ψ τv = xψ w Λ π e ( i π ) y π e π ψ τv = X u ∈ S d ψ u f u Λ π e ( i π ) y π e π ψ τv ⊆ I π by Theorem 2.3 for the second equality and Lemma 5.15 for the inclusion. (cid:3) We can now complete the proof of Theorem 5.6. We have already provedthat I π is an ideal containing e ( i π ). By definition, I π contains I >π which byinduction is equal to P σ>π R α e ( i σ ) R α . Therefore I π ⊇ P σ ≥ π R α e ( i σ ) R α .On the other hand, I π is spanned by I >π and elements from ψ w y π Λ π e π ψ τv ⊆ R α e ( i π ) R α ( w, v ∈ S π ) . Therefore I π ⊆ P σ ≥ π R α e ( i σ ) R α , and so we have equality. This concludesthe proof of Theorem 5.6.5.4. Affine cellular basis.
The following lemma shows that the cell idealsexhaust the algebra R α . Lemma 5.20.
If a two-sided ideal J of R α contains all idempotents e ( i π ) with π ∈ Π( α ) , then J = R α . In particular, P π ∈ Π( α ) I ′ π = R α .Proof. If J = R α , let I be a maximal (graded) left ideal containing J .Then R α /I ∼ = L ( π ) for some π . Then e ( i π ) L ( π ) = 0, which contradicts theassumption that e ( i π ) ∈ J . This argument proves the lemma over any field,and then it also follows for Z . (cid:3) Theorem 5.21.
For each π ∈ Π( α ) pick a Z -basis { b π,x } x ∈ X ( π ) of Λ π . Then { ψ w y π b π,x e π ψ τv | π ∈ Π( α ) , x ∈ X ( π ) , v, w ∈ S π } (5.22) is a Z -basis of R α . In particular, R α = L π ∈ Π( α ) I ′ π .Proof. The elements of (5.22) span R α in view of Lemma 5.20. So it remainsto apply Proposition 3.5. (cid:3) Theorem 5.23.
Let π ∈ Π( α ) , and ¯ R α := R α /I >π . Then (i) the map Λ π → ¯ e π ¯ R α ¯ e π , b ¯ b ¯ e π is an isomorphism of graded alge-bras; (ii) ¯ R α ¯ e π is a free right ¯ e π ¯ R α ¯ e π -module with basis { ¯ ψ w ¯ y π ¯ e π | w ∈ S π } ; (iii) ¯ e π ¯ R α is a free left ¯ e π ¯ R α ¯ e π -module with basis { ¯ e π ¯ ψ τv | v ∈ S π } ; (iv) multiplication provides an isomorphism ¯ R α ¯ e π ⊗ ¯ e π ¯ R α ¯ e π ¯ e π ¯ R α ∼ −→ ¯ R α ¯ e π ¯ R α ;(v) ¯ R α ¯ e π ¯ R α = I π /I >π .Proof. (v) follows from Theorem 5.6.Now pick a Z -basis { b x } x ∈ X of Λ π . We next prove that { ¯ ψ w ¯ y π ¯ b x ¯ e π | w ∈ S π , x ∈ X } (5.24)is a Z -basis of ¯ R α ¯ e π and { ¯ ψ π ¯ y π ¯ b x ¯ e π ¯ ψ τv | v ∈ S π , x ∈ X } (5.25)is a Z -basis of ¯ e π ¯ R α .To prove that (5.24) is a basis of ¯ R α ¯ e π , note that ¯ R α ¯ e π ¯ R α = ¯ I π = ¯ I ′ π by (v), so ¯ R α ¯ e π = ¯ I ′ π ¯ e π . A Z -spanning set for ¯ R α ¯ e π is therefore given by¯ ψ w ¯ y π ¯ b x ¯ e π ¯ ψ τv ¯ e π where v, w ∈ S π . Using Lemma 5.3, we have ¯ e π ¯ ψ τv ¯ e π = 0unless v is a block permutation. On the other hand, if v is a nontrivial blockpermutation, then ¯ e π ¯ ψ τv ¯ e π = ¯ e π ¯ ψ τv ¯ ψ π ¯ y π ¯ e ( i π ) = 0 in view of Lemma 5.13.We can therefore refine our spanning set to (5.24), which is Z -linearly inde-pendent by Theorem 5.21.To prove that (5.25) is a basis of ¯ e π ¯ R α , note that by definition we have¯ e π ¯ R α = ¯ ψ π ¯ y π ¯ e ( i π ) ¯ R α ⊆ ¯ ψ π ¯ e ( i π ) ¯ R α . On the other hand, by (4.10) andLemma 5.13, we have ¯ ψ π ¯ e ( i π ) = ¯ e π ¯ ψ π ¯ e ( i π ), hence ¯ ψ π ¯ e ( i π ) ¯ R α ⊆ ¯ e π ¯ R α . Thus¯ ψ π ¯ e ( i π ) ¯ R α = ¯ e π ¯ R α . A spanning set for ¯ ψ π ¯ e ( i π ) ¯ R α is given by¯ ψ π ¯ e ( i π ) ¯ ψ w ¯ y π ¯ b x ¯ e π ¯ ψ τv for v, w ∈ S π and x ∈ X . As in the previous paragraph, this term is zerofor w = 1. The spanning set thus reduces to the elements of (5.25), whichare linearly independent by Theorem 5.21.(i) We have ¯ e π ¯ R α ¯ e π = ¯ e π ¯ R α T ¯ R α ¯ e π , and both of the sets on the right arespanned by a subset of our basis (5.22) of ¯ R α . Hence ¯ e π ¯ R α T ¯ R α ¯ e π has abasis given by those basis elements in both (5.24) and (5.25). This is clearlythe set of all ¯ ψ π ¯ y π ¯ b x ¯ e π = ¯ e π ¯ b x ¯ e π = ¯ b x ¯ e π for x ∈ X . The mapΛ π → ¯ e π ¯ R α ¯ e π b ¯ e π ¯ b ¯ e π = ¯ b ¯ e π is therefore an isomorphism.(ii) follows immediately from (i) and the fact that (5.24) is a basis of¯ R α ¯ e π .(iii) follows from (i), the fact that (5.25) is a basis of ¯ e π ¯ R α , and theequality ¯ ψ π ¯ y π ¯ b x ¯ e π ¯ ψ τv = ¯ b x ¯ e π ¯ e π ¯ ψ τv = ¯ b x ¯ e π ¯ ψ τv . (iv) follows immediately by considering the bases constructed above. (cid:3) Recall the definition of an affine cellular algebra given in the introduction.
Corollary 5.26.
The algebra R α is affine cellular with cell chain given bythe ideals { I π | π ∈ Π( α ) } . FFINE CELLULARITY OF KLR ALGEBRAS IN TYPE A 19
Proof.
Fix π ∈ Π( α ) and write ¯ R α := R α /I >π . We must show that I π /I >π is an affine cell ideal of ¯ R α . We verify conditions (i)–(iii) of the definitiongiven in the introduction.(i) This follows from Lemma 5.5.(ii) Define V to be the free Z -module on the basis { w | w ∈ S π } , andlet B := Λ π with σ being the trivial involution of B . We define the right B -module ∆ := V ⊗ Z B . Theorem 5.23(i),(ii) implies that the map∆ → ¯ R α ¯ e π , w ⊗ b ¯ ψ w ¯ y π ¯ e π ¯ b where w ∈ S π , b ∈ B , is an isomorphism of right B -modules. We use thisto define an ¯ R α - B -bimodule structure on ∆.(iii) Let ∆ ′ := B ⊗ Z V be the B - ¯ R α -bimodule defined as in (1.1). ByTheorem 5.23(i),(iii), the map∆ ′ → ¯ e π ¯ R α , b ⊗ w ¯ b ¯ e π ¯ ψ τw (5.27)for w ∈ S π , b ∈ B , defines an isomorphism of left B -modules. Claim.
The map (5.27) is an isomorphism of B - ¯ R α -bimodules. Proof of Claim.
Note that under the identifications ∆ ≃ ¯ R α ¯ e π and ∆ ′ ≃ ¯ e π ¯ R α (as B -modules), the twist map s − : ∆ ′ → ∆ becomes the map η : ¯ e π ¯ R α → ¯ R α ¯ e π , ¯ b ¯ e π ¯ ψ τw ¯ ψ w ¯ y π ¯ e π ¯ b. Therefore the claim is equivalent to η (¯ b ¯ e π ¯ ψ τw r ) = r τ ¯ ψ w ¯ y π ¯ e π ¯ b for w ∈ S π , b ∈ B , and r ∈ ¯ R α . We can write¯ b ¯ e π ¯ ψ τw r = X v ∈ S π ¯ b v ¯ e π ¯ ψ τv for some b v ∈ Λ π , and then, using (5.1), we get η (¯ b ¯ e π ¯ ψ τw r ) = η (cid:16) X v ∈ S π ¯ b v ¯ e π ¯ ψ τv (cid:17) = X v ∈ S π ¯ ψ v ¯ y π ¯ e π ¯ b v = X v ∈ S π ¯ ψ v ¯ y π ¯ ψ π ¯ y π ¯ e ( i π )¯ b v = X v ∈ S π ¯ ψ v ¯ e ( i π )¯ y π ¯ ψ π ¯ b v ¯ y π = X v ∈ S π (cid:0) ¯ b v ¯ ψ π ¯ y π ¯ e ( i π ) ¯ ψ τv (cid:1) τ ¯ y π = X v ∈ S π (cid:0) ¯ b v ¯ e π ¯ ψ τv (cid:1) τ ¯ y π = (cid:0) ¯ b ¯ e π ¯ ψ τw r (cid:1) τ ¯ y π = (cid:0) ¯ b ¯ ψ π ¯ y π ¯ e ( i π ) ¯ ψ τw r (cid:1) τ ¯ y π = r τ ¯ ψ w ¯ e ( i π )¯ y π ¯ ψ π ¯ b ¯ y π = r τ ¯ ψ w ¯ y π ¯ e π ¯ b. The proof of the claim is now complete, and so we can identify ∆ ′ with¯ e π ¯ R α .By Theorem 5.23(iv), the map ∆ ⊗ B ∆ ′ → ¯ I π is an isomorphism. Oneshows that the diagram in part (iii) of the definition of a cell ideal is com-mutative using an argument similar to the one in the proof of the aboveclaim. (cid:3) References [1] J. Brundan and A. Kleshchev, Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras,
Invent. Math. (2009), 451–484.[2] J. Brundan and A. Kleshchev, Graded decomposition numbers for cyclotomicHecke algebras,
Adv. Math. (2009), 1883–1942[3] J. Brundan, A. Kleshchev, and W. Wang, Graded Specht modules,
J. reine angew.Math. , (2011), 61–87. [4] R. Dipper, G. D. James and A. Mathas, Cyclotomic q -Schur algebras, Math. Z. (1998), 385–416.[5] W. Fulton,
Young Tableaux , LMS Students Text Studies 35, CUP, Cambridge,1997.[6] J. Hu and A. Mathas, Graded cellular bases for the cyclotomic Khovanov-Lauda-Rouquier algebras of type A,
Adv. Math. , (2010), 598–642.[7] J. Humphreys, Reflection Groups and Coxeter Groups , Cambridge Studies in Ad-vanced Mathematics 29, CUP, Cambridge, 1990.[8] V. G. Kac,
Infinite Dimensional Lie Algebras , Cambridge University Press, 1990.[9] S. Kato, PBW bases and KLR algebras, arXiv:1203.5254 .[10] M. Khovanov and A. Lauda, A diagrammatic approach to categorification of quan-tum groups I,
Represent. Theory (2009), 309–347.[11] A. Kleshchev and J. W. Loubert, Affine cellularity of Khovanov-Lauda-Rouquieralgebras in ADE types, in preparation .[12] A. Kleshchev, A. Mathas, and A. Ram, Universal graded Specht modules for cy-clotomic Hecke algebras, Proc. Lond. Math. Soc. , to appear; arXiv:1102.3519 .[13] A. Kleshchev and A. Ram, Representations of Khovanov-Lauda-Rouquier algebrasand combinatorics of Lyndon words,
Math. Ann. (2011), no. 4, 943–975.[14] S. Koenig and C. Xi, Affine cellular algebras,
Adv. Math. (2012), no. 1, 139–182.[15] L. Manivel,
Symmetric functions, Schubert polynomials and Degeneracy Loci , vol.6 of SMF/AMS Texts and Monographs. AMS, Providence, RI, 2001.[16] P.J. McNamara, Finite dimensional representations of Khovanov-Lauda-Rouquieralgebras I: finite type, arXiv:1207.5860 .[17] R. Rouquier, 2-Kac-Moody algebras; arXiv:0812.5023 . Department of Mathematics, University of Oregon, Eugene, OR 97403, USA
E-mail address : [email protected] Department of Mathematics, University of Oregon, Eugene, OR 97403, USA
E-mail address : [email protected] School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, UK
E-mail address ::