Cellularity of generalized Schur algebras via Cauchy decomposition
aa r X i v : . [ m a t h . R T ] F e b CELLULARITY OF GENERALIZED SCHUR ALGEBRASVIA CAUCHY DECOMPOSITION
JONATHAN D. AXTELL
Abstract.
We describe a generalization of Hashimoto and Kurano’s Cauchyfiltration for divided powers algebras. This filtration is then used to providea cellular structure for generalized Schur algebras associated to an arbitrarycellular algebra, A . Applications to the cellularity of wreath product algebras A ≀ S d are also considered. Introduction
Let k be a noetherian integral domain and suppose A is a cellular k -algebra [8].Then Geetha and Goodman [7] showed that the wreath product algebra A ≀ S d = A ⊗ d ⋊ k S d is cellular, provided that all of the cell ideals of A are cyclic. On the other hand, thegeneralized Schur algebras S A ( n, d ) were defined by Evseev and Kleshchev [4, 5] inorder to prove the Turner double conjecture. These algebras are related to wreathproduct algebras by a generalized Schur-Weyl duality established in [4].In this paper, we describe a cellular structure for the generalized Schur algebra S A ( n, d ) for an arbitrary cellular algebra A and for all integers n, d ≥
0. Thisextends some results of Kleshchev and Muth [13, 14, 15]. It follows, for example,from results of [15] that the algebra S A ( n, d ) is cellular for certain algebras A whichare both cellular and quasi-hereditary. We note that for such algebras, the cell idealsare automatically cyclic. The method used in this paper, however, does not requireany additional assumptions on the cellular algebra.Our approach is motivated by that of [17], where Krause used the Cauchy de-composition of divided powers [1, 12] to describe the highest weight structure ofcategories of strict polynomial functors. As Krause mentions, this leads to an al-ternate proof of the fact that classical Schur algebras S k ( n, d ) are quasi-hereditary,which follows by a Morita equivalence. As we will see, this approach can similarlybe used to describe cellular structure.We begin by constructing a generalized Cauchy filtration for the divided powersΓ d J of a given k -module, J , which we assume is equipped with a filtration0 = J ⊂ · · · ⊂ J r = J such that J j /J j − ∼ = U j ⊗ k V j , for some free k -modules U j , V j of finite rank. Ourfirst main result is a generalized Cauchy decomposition formula (Theorem 5.14),which provides a filtration of Γ d J such that the associated graded object is a directsum of modules of the form M λ ∈ Λ U λ ⊗ k V λ , where U λ , V λ are generalized Weyl modules defined in Section 5.6 and Λ denotes aset of r -multipartitions. This paper was supported by the National Research Foundation of Korea (NRF) funded bythe Ministry of Science (NRF-2017R1C1B5018384).
The generalized Schur algebra S A ( n, d ) may be identified as the d -th dividedpower Γ d M n ( A ), where M n ( A ) is the algebra of size n matrices over A . We are thusable to use the above decomposition, together with K¨onig and Xi’s characterizationof cellular algebras in [16], to prove our second main result (Theorem 6.5) whichshows that generalized Schur algebras are cellular. In Example 6.6, we describe acorresponding cellular basis explicitly for a particular case, S Z (1 , Z is azig-zag algebra (considered as an ordinary algebra rather than a superalgebra, asin [15]).As a consequence of generalized Schur-Weyl duality, Corollary 6.8 shows thatthe wreath product algebras A ≀ S d are cellular for an arbitrary cellular algebra A .This provides an alternate proof of the main result in [7], for the case where A iscyclic cellular, and a more recent result of Green [11], for the general case where A is an arbitrary cellular algebra. 2. Preliminaries
Assume throughout that k is a commutative ring, unless mentioned otherwise.The notation ♯ is used for the cardinality of a set.2.1. Weights, partitions, and sequences.
Write N and N to denote the setsof positive and nonnegative integers, respectively, with the usual total order. Moregenerally, suppose that B is a countable totally ordered set which is bounded below.Any elements a, b ∈ B determine an interval[ a, b ] := { c ∈ B | a ≤ c ≤ b } which is empty unless a ≤ b .A weight ( on B ) is a sequence of nonnegative integers µ = ( µ b ) b ∈ B such that µ b = 0 for almost all b . Let Λ( B ) denote the set of all weights on B . A partition ( on B ) is a weight λ ∈ Λ( B ) such that b < c implies λ b ≥ λ c , ∀ b, c ∈ B . The subset of partitions is denoted Λ + ( B ) ⊂ Λ( B ). The size of a weight µ is theinteger | µ | := P b µ b . Let Λ d ( B ) denote the set of all weights of size d and writeΛ + d ( B ) := Λ + ( B ) ∩ Λ d ( B )for each d ∈ N . Remark 2.1.
In this notation and elsewhere, we will use the convention of replac-ing an argument of the form [1 , n ] by “ n ” for any n ∈ N , so that for example Λ( n )denotes the set Λ([1 , n ]) of weights of the form µ = ( µ , . . . , µ n ).We also identify each set Λ( n ) as a subset of Λ( N ) in the obvious way and write l ( µ ) := min { n ∈ N | µ ∈ Λ( n ) } to denote the length of a weight µ ∈ Λ( N ). For example, the length l ( λ ) of apartition λ = ( λ , λ , . . . ) in Λ + ( N ) equals the number of positive parts, λ i ∈ N . Definition 2.2.
Let d ∈ N . Recall that the lexicographic ordering on Λ d ( N ) is thetotal order defined by setting λ ≤ µ if λ j ≤ µ j whenever λ i = µ i for all i < j . Weuse the notation (cid:22) to denote the restriction of ≤ to the subset Λ + d ( N ) of partitionsof size d .Now fix d ∈ N , and write seq d ( B ) to denote the set of all functions b : [1 , d ] → B . ELLULARITY OF GENERALIZED SCHUR ALGEBRAS 3
We identify seq d ( B ) with B d by setting b = ( b , . . . , b d ), with b i = b ( i ) for all i ∈ [1 , d ]. The symmetric group S d of permutations of [1 , d ] acts on seq d ( B ) fromthe right via composition. We write b ∼ c if there exists σ ∈ S d with c = b σ .The weight of a sequence b ∈ seq d ( B ) is the element of Λ d ( B ) defined by µ ( b ) := ( µ c ) c ∈ B , where µ c = ♯ { i | b i = c } ∀ c ∈ B . We note the following elementary result.
Lemma 2.3.
The map µ : seq d ( B ) → Λ d ( B ) , sending b µ ( b ) , induces a bijec-tion: seq d ( B ) / S d ≃ Λ d ( B ) . Proof.
We may assume that B is nonempty. Since B is bounded below, it is possibleto write the elements explicity in the form B = { b B < b B < . . . } . (2.1)To show that the map b µ ( b ) is surjective, note that a right inverse is given byΛ d ( B ) → seq d ( B ) : µ b µ := ( b B , . . . , b B , b B , . . . , b B , . . . )where b B occurs with multiplicity µ b B , etc. Finally, it is easy to see that b ∼ c ifand only if µ ( b ) = µ ( c ), which completes the proof. (cid:3) Suppose more generally that B , . . . , B r is a collection of bounded below, totallyordered sets. We again consider the product B = B × · · ·× B r as a bounded below,totally ordered set via the lexicographic ordering.The symmetric group S d acts diagonally on the following product seq d ( B , . . . , B r ) := seq d ( B ) × · · · × seq d ( B r ) . Notice that the bijection θ : seq d ( B , . . . , B r ) ≃ seq d ( B )defined by θ ( b (1) , . . . , b ( r ) ) : i ( b (1) i , . . . , b ( r ) i ) , ∀ i ∈ [1 , d ] , is S d -equivariant. It thus follows as an immediate consequence of Lemma 2.3 thatthere is a bijection seq d ( B , . . . , B r ) / S d ≃ Λ d ( B ) , (2.2)where seq d ( B , . . . , B r ) / S d denotes the set of diagonal S d -orbits.2.2. Multipartitions.
Suppose d ∈ N and let B , . . . , B r be as above. Then weuse the following notation for the productΛ + ( B , . . . , B r ) := Λ + ( B ) × · · · × Λ + ( B r ) . whose elements are called r -multipartions and denoted λ = ( λ (1) , . . . , λ ( r ) ). The weight of an r -multipartion λ is the element of Λ( r ) defined by | λ | := ( | λ (1) | , . . . , | λ ( r ) | ) . We call || λ || := P | λ ( j ) | the total weight (or size ) of λ .Given µ ∈ Λ( r ) and d ∈ N , we writeΛ + µ ( B , . . . , B r ) := Λ + µ ( B ) × · · · × Λ + µ r ( B r )and Λ + d ( B , . . . , B r ) := G ν ∈ Λ d ( r ) Λ + ν ( B , . . . , B r )to denote the subset of r -multipartions of weight µ , resp. total weight d .In the special case where B j = N for j ∈ [1 , r ], note thatΛ + ( N , . . . , N ) = Λ + ( N ) r . JONATHAN D. AXTELL
We then use the following notationΛ + d ( N ) r := Λ + d ( N , . . . , N ) , Λ + µ ( N ) r := Λ + µ ( N , . . . , N )for d ∈ N and µ ∈ Λ r ( d ), respectively.The next definition describes a total order on the set of r -multipartitions of afixed total weight. Definition 2.4.
Suppose d, r ∈ N . Then Λ + d ( N ) r has a total order (cid:22) defined asfollows. For r -multipartitions µ , λ ∈ Λ + ν ( N ) of weight ν ∈ Λ d ( r ), we set λ (cid:22) µ if λ ( j ) (cid:22) µ ( j ) , whenever λ ( i ) = µ ( i ) for all i < j. We then extend (cid:22) to all of Λ + d ( N ) r by setting λ ≺ µ whenever | λ | < | µ | in thelexicographic ordering on Λ d ( r ).Suppose n , . . . , n r ∈ N and d ∈ N . Recalling the notation from Remark 2.1,we identify the set of r -multipartionsΛ + ( n , . . . , n r ) := Λ + ([1 , n ] , . . . , [1 , n r ])as a subset of Λ + ( N ) r and view (cid:22) as a total order on Λ + d ( n , . . . , n r ) by restriction.2.3. Finitely generated projective modules.
Let M k denote the category of all k -modules and k -linear maps. The full subcategory of finitely generated projective k -modules is denoted P k .Given M, N ∈ M k , we write M ⊗ N = M ⊗ k N and Hom( M, N ) = Hom k ( M, N ).Also write End( M ) to denote the k -algebra Hom( M, M ). If M ∈ P k , we let M ∨ =Hom( M, k ) denote the k -linear dual. For any M, M ′ , N, N ′ ∈ P k , there is anisomorphism Hom( M ⊗ N, M ′ ⊗ N ′ ) ∼ = Hom( M, M ′ ) ⊗ Hom(
N, N ′ ) (2.3)which is natural with respect to composition.2.4. Divided and symmetric powers.
Let d ∈ N . Given M ∈ P k , there is a rightaction of the symmetric group S d on the tensor power M ⊗ d given by permutingtensor factors. We define the d -th divided power of M to be the invariant submoduleΓ d M := ( M ⊗ d ) S d . Similarly, the coinvariant module is denotedSym d M := ( M ⊗ d ) S d and called the d -th symmetric power of M . It follows by definition thatΓ d ( M ) ∨ ∼ = Sym d ( M ∨ ) . (2.4)We also set Γ M = Sym M = k .Note that the isomorphism (2.4) is usually taken as the definition of Γ d M (cf.[1]), while we have used the equivalent definition from [17] in terms of symmetrictensors.2.5. The divided powers algebra.
The category M k (resp. P k ) is a symmetricmonoidal category with symmetry isomorphism tw : M ⊗ N ∼ −→ N ⊗ M (2.5)defined by x ⊗ y y ⊗ x , for all x ∈ M, y ∈ N .Suppose M ∈ P k . Then Γ( M ) := M d ∈ N Γ d M ELLULARITY OF GENERALIZED SCHUR ALGEBRAS 5 is an ( N -graded) commutative algebra called the divided powers algebra , withmultiplication defined on homogeneous components via the shuffle product: for x ∈ Γ d M and y ∈ Γ e M , define x ∗ y := X σ ∈ S d,ed + e ( x ⊗ y ) σ where S d,ed + e is the quotient group S d + e / S d × S e . For example, we have x ⊗ d ∗ x ⊗ e = (cid:0) d + ed (cid:1) x ⊗ ( d + e ) for any x ∈ M .There is also a comultiplication, ∆ : Γ( M ) → Γ( M ) ⊗ Γ( M ), which is the N -homogenous map whose graded components∆ : Γ d M → Γ d − c M ⊗ Γ c M are defined as the inclusions( M ⊗ d ) S d ֒ → ( M ⊗ d ) S d − c × S c induced by the embeddings S d − c × S c ֒ → S d , for c ∈ [0 , d ]. These maps, togetherwith the unit, k = Γ M ֒ → Γ( M ), and the counit, Γ( M ) ։ Γ M (projection ontodegree 0), make Γ( M ) into a bialgebra.2.6. Decompositions.
The symmetric algebra S ( M ) is defined as the free com-mutative k -algebra generated by M and has a decomposition S ( M ) = M d ∈ N Sym d M. It follows that S ( − ) defines a functor from P k to the category of all commutative k -algebras, which preserves coproducts. Hence S ( M ) ⊗ S ( N ) ∼ = S ( M ⊕ N ), and bythe duality (2.4) there is an isomorphismΓ( M ) ⊗ Γ( N ) ≃ Γ( M ⊕ N ) . (2.6)The isomorphism (2.6) is given explicitly by restricting the multiplication map x ⊗ y x ∗ y , where Γ( M ), Γ( N ) are considered as subalgebras of Γ( M ⊕ N ). Itfollows that for each d ∈ N there is a decompositionΓ d ( M ⊕ N ) = M ≤ c ≤ d Γ c ( M ) ∗ Γ d − c ( N ) (2.7)where Γ c ( M ) ∗ Γ d − c ( N ) denotes the image of Γ c ( M ) ⊗ Γ d − c ( N ) under (2.6).Note that Γ d k ∼ = k for all d ∈ N . Thus, given a free k -module V of finite rank, itfollows by induction from (2.7) that the divided power Γ d V is again a free k -moduleof finite rank. For example, suppose V has a finite ordered k -basis { x b } b ∈ B . ThenΓ d V has the following k -basis n x µ := Y b ∈ B x ⊗ µ b b | µ ∈ Λ d ( B ) o (2.8)where the product denotes multiplication in Γ( V ).The basis (2.8) can also be parameterized by elements of seq d ( B ). First noticethat the the tensor power V ⊗ d has the following basis { x ⊗ b := x b ⊗ . . . ⊗ x b d (cid:12)(cid:12)(cid:12) b ∈ seq d ( B ) } . Given b ∈ seq d ( B ), we then define x b := P b ∼ c x ⊗ c . Notice that x b = x µ ( b ) . Itthen follows from Lemma 2.3 that the set { x b | b ∈ seq d ( B ) / S d } (2.9)is also a basis of Γ d V , indexed by any complete set of orbit representatives. JONATHAN D. AXTELL
Polynomial functors.
We recall the definitions of some well known polyno-mial endofunctors on the category P k along with their associated natural transfor-mations.Let d ∈ N . Then recall the functor ⊗ d : P k → P k sending M M ⊗ d , whoseaction on morphisms is defined by ⊗ dM,N ( ϕ ) := ϕ ⊗ · · · ⊗ ϕ : M ⊗ d → N ⊗ d for any ϕ ∈ Hom(
M, N ).It follows easily from (2.7) that the divided power Γ d M of a finitely-generated,projective k -module M ∈ P k is again finitely-generated and projective. This yieldsa functor Γ d : P k → P k which is a subfunctor of ⊗ d . In particular, the action of Γ d on morphisms is defined by restrictionΓ dM,N ( ϕ ) := ( ϕ ⊗ d ) | Γ d M : Γ d M → Γ d N for any ϕ ∈ Hom(
M, N ).Now let
S, T : P k → P k be an arbitrary pair of functors. Then the tensor product − ⊗ − induces the following bifunctors S ⊠ T, T ( − ⊗ − ) : P k × P k → P k which are respectively defined by S ⊠ T := ( − ⊗ − ) ◦ ( S × T ) , T ( − ⊗ − ) := T ◦ ( − ⊗ − ) . We also have the “object-wise” tensor product S ⊗ T : P k → P k defined by S ⊗ T := ( S ⊠ T ) ◦ δ (2.10)where δ : P k → P k × P k denotes the diagonal embedding: M ( M, M ).Now suppose
M, N ∈ P k . As in [17], define ψ d = ψ d ( M, N ) to be the uniquemap which makes the following square commute:Γ d M ⊗ Γ d N Γ d ( M ⊗ N ) M ⊗ d ⊗ N ⊗ d ( M ⊗ N ) ⊗ dψ d ∼ (2.11)The following lemma is easy to check. Lemma 2.5. (1) The maps ψ d ( M, N ) form a natural transformation of bi-functors ψ d : Γ d ⊠ Γ d → Γ d ( − ⊗ − ) . (2) If M, N ∈ P k , then the following diagram commutes Γ d M ⊗ Γ d N Γ d ( M ⊗ N )Γ d N ⊗ Γ d M Γ d ( N ⊗ M ) ψ d ( M,N ) tw Γ d ( tw ) ψ d ( N,M ) where tw permutes tensor factors as in (2.5). Generalized Schur Algebras
After recalling the definition of generalized Schur algebras [4] associated to a k -algebra A , we introduce corresponding standard homomorphisms between certainmodules of divided powers. ELLULARITY OF GENERALIZED SCHUR ALGEBRAS 7
Associative k -algebras. Suppose that
R, S are associative algebras in thecategory M k . Recall that the tensor product R ⊗ S is the algebra in M k withmultiplication m R ⊗ S defined by R ⊗ S ⊗ R ⊗ S ⊗ tw ⊗ −−−−−→ R ⊗ R ⊗ S ⊗ S m R ⊗ m S −−−−−−→ R ⊗ S. Given d ∈ N , the tensor power R ⊗ d is an associative algebra in M k in a similar way.If R is unital, then R ⊗ d has unit 1 ⊗ dR .In the remainder, the term k -algebra will always refer to a unital, associativealgebra in the category P k . Let A ∈ P k be a k -algebra. Then A -mod (resp. mod- A )denotes the subcategory of P k consisting of all left (right) A -modules, M ∈ P k ,and A -module homomorphisms. Write Hom A ( M, N ) ∈ P k to denote the set of all A -homomorphisms from M to N for M, N ∈ A -mod (resp. mod- A ). We also write ρ M : A ⊗ M → A (resp. ρ M : M ⊗ A → A ) to denote the induced linear mapcorresponding to a left (right) A -module.If M ∈ A -mod (resp. mod- A ) and N ∈ B -mod (resp. mod- B ), the tensor product M ⊗ N is a left (resp. right) A ⊗ B -module, with corresponding module map: ρ M ⊗ N = ( ρ M ⊗ ρ N ) ◦ (1 ⊗ τ ⊗ The algebra Γ d A . Suppose A is a k -algebra. Then Γ d A is a k -algebra withmultiplication m Γ d A defined via the compositionΓ d A ⊗ Γ d A ψ d −−→ Γ d ( A ⊗ A ) Γ d ( m A ) −−−−−→ Γ d A, where the second map denotes the functorial action of Γ d on m A . It follows thatΓ d A is a unital subalgebra of A ⊗ d . Example 3.1 (The Schur algebra) . Suppose n ∈ N , and let M n ( k ) denote thealgebra of all n × n -matrices in k . Then Γ d M n ( k ) is isomorphic to the classical Schuralgebra , S ( n, d ), defined by Green [9, Theorem 2.6c]. We view this isomorphism asan identification.We now have two distinct multiplications on the direct sum Γ( A ) = L d ∈ N Γ d A .In order to distinguish them, we sometimes refer to the shuffle product ∇ : Γ d A ⊗ Γ e A → Γ d + e A : x ⊗ y x ∗ y as outer multiplication in Γ( A ), while inner multiplication refers to the map definedas multiplication in Γ d A on diagonal components m Γ d A : Γ d A ⊗ Γ d A → Γ d A : x ⊗ y xy and then extended by zero to other components.3.3. Generalized Schur algebras.
Given a k -algebra A , write M n ( A ) for thealgebra of n × n -matrices in A . We identify M n ( A ) with M n ( k ) ⊗ A viaM n ( A ) ∼ −→ M n ( k ) ⊗ A : ( a ij ) X i,j E ij ⊗ a ij , where E ij are elementary matrices in M n ( k ). Next, suppose V is any left (resp. right)M n ( k )-module, and let M ∈ A -mod (mod- A ). Then write V ( M ) := V ⊗ M to de-note the corresponding M n ( A )-module. Definition 3.2.
Suppose A is an algebra, and let n ∈ N , d ∈ N . Then the generalized Schur algebra S A ( n, d ) is the algebra Γ d M n ( A ).Using the notation of [4], notice that M n is spanned by the elements ξ ai,j := E ij ⊗ a , for all a ∈ A and i, j ∈ [1 , n ]. Now suppose that A is free as a k -modulewith finite ordered basis { x b } b ∈ B . Then M n ( A ) has a corresponding basis { ξ i,j,b := ξ x b i,j | i, j ∈ [1 , n ] , b ∈ B } . JONATHAN D. AXTELL
We view M n ( k ) as a subalgebra of M n ( A ) by identifying E ij = ξ i,j . Notice thatthe classical Schur algebra S ( n, d ) is thus a (unital) subalgebra of S A ( n, d ).For each triple ( i , j , b ) ∈ seq d ( n, n, B ), there is a corresponding element of S A ( n, d ) denoted by ξ i , j , b := X ( i , j , b ) ∼ ( r , s , c ) ξ r ,s ,c ⊗ · · · ⊗ ξ r d ,s d ,c d , where the sum is over all triples ( r , s , c ) in the same diagonal S d -orbit as ( i , j , b ).It thus follows from (2.2), (2.8) and (2.9) that the set { ξ i , j , b | ( i , j , b ) ∈ seq d ( n, n, B ) / S d } forms a basis of of S A ( n, d ). In a similar way, the subalgebra S ( n, d ) has a basisgiven by { ξ i , j := X ( i , j ) ∼ ( r , s ) ξ r ,s ⊗ · · · ⊗ ξ r d ,s d | ( i , j ) ∈ seq d ( n, n ) / S d } . For each weight µ ∈ Λ d ( n ), we write ξ µ := ξ i µ , i µ to denote the corresponding idempotent in S ( n, d ) ⊂ S A ( n, d ).3.4. Standard homomorphisms.
Let us fix an algebra A throughout the remain-der of the section. Given M ∈ A -mod, it follows from (2.11) that Γ d M is a leftΓ d A -module with module map ρ Γ d M determined by the compositionΓ d A ⊗ Γ d M ψ d −−→ Γ d ( A ⊗ M ) Γ d ( ρ M ) −−−−−→ Γ d ( M ) , where the second map denotes the functorial action of Γ d on ρ M . Lemma 3.3.
Suppose
M, N ∈ A -mod, and let ϕ : M → N be an A -module homo-morphism. Then the functorial map Γ d ( ϕ ) : Γ d M → Γ d N is a homomorphism of Γ d A -modules. Moreover, if ϕ is injective (resp. surjective)then so is Γ d ( ϕ ) .Proof. The map ϕ ⊗ d : M ⊗ d → N ⊗ d is a homomorphism of A ⊗ d -modules, and if ϕ is injective (resp. surjective) then so is ϕ ⊗ d . The statements for Γ d ( ϕ ) follow byrestriction. (cid:3) Suppose d, e ∈ N and M, N ∈ A -mod. Notice that the homogeneous componentof comultiplication ∆ : Γ d + e A → Γ d A ⊗ Γ e A (3.1)is an injective (unital) map of k -algebras. It follows that Γ d M ⊗ Γ e N has a corre-sponding Γ d A -module structure, defined by restriction along (3.1). In the particularcase M = N , we note that each of the following maps is a Γ d A -module homomor-phism: ∆ : Γ d + e M → Γ d M ⊗ Γ e M, ∇ : Γ d M ⊗ Γ e M → Γ d + e M, tw : Γ d M ⊗ Γ e M ∼ −→ Γ e M ⊗ Γ d M, (3.2)where ∇ (resp. ∆) are components of (co)multiplication in the bialgebra Γ( M ).Setting A = k then gives the following. ELLULARITY OF GENERALIZED SCHUR ALGEBRAS 9
Lemma 3.4.
Let d, e ∈ N . Then there are natural transformations ∆ : Γ d + e → Γ d ⊗ Γ e , ∇ : Γ d ⊗ Γ e → Γ d + e of functors P k → P k induced by setting ∆( M ) (resp. ∇ ( M ) ) equal to (co)multiplicationin Γ( M ) , for each M ∈ P k . Now suppose r ∈ N and µ ∈ Λ( r ). Given M, N , . . . , N r ∈ P k , we writeΓ ( µ ) ( N , . . . , N r ) := Γ µ N ⊗ · · · ⊗ Γ µ r N r and set Γ µ M := Γ ( µ ) ( M, . . . , M ) . If M , . . . , M r ∈ A -mod, then we consider Γ ( µ ) ( M , . . . , M r ) as a left Γ d A -moduleby restriction along the corresponding inclusion, ∆ : Γ d A → Γ µ A , of k -algebras.Suppose that γ = ( γ ij ) ∈ Λ d ( N × N ) is a (semi-infinite) matrix whose entries sumto d . Then let λ, µ ∈ Λ d ( N ) be weights such that λ i = P j γ ij and µ j = P i γ ij forall i, j ∈ N . Slightly abusing notation, for a given N ∈ P k , we also write γ = γ ( N )to denote the corresponding standard homomorphism: γ : Γ µ N → Γ λ N defined by the composition O j Γ µ j N ∆ ⊗ ... ⊗ ∆ −−−−−−→ O i O j Γ γ ij N ∼ −→ O j O i Γ γ ij N ∇⊗ ... ⊗∇ −−−−−−→ O i Γ λ i N, where each ∇ (resp. ∆) denotes an appropriate component of (co)multiplication inthe bialgebra Γ( N ), and where the second map rearranges the tensor factors.If M ∈ A -mod, then it follows from (3.2) that γ ( M ) : Γ µ M → Γ λ M is ahomomorphism of Γ d A -modules. In the same way, we obtain homomorphisms of S A ( n, d )-modules corresponding to any given M ∈ M n ( A )-mod.3.5. Quotient modules.
Suppose M ∈ P k . Then we write h L i ⊂ M ⊗ d to denotethe S d -submodule generated by a subset L ⊂ M ⊗ d . For example if L , . . . , L d ⊂ M are k -submodules and L = L ⊗ · · · ⊗ L d , then h L i = X σ ∈ S d L σ ⊗ · · · ⊗ L dσ , where iσ := σ − ( i ) denotes the right action of σ on i ∈ [1 , d ].Now suppose M = N ⊕ N ′ for some k -submodules N, N ′ ⊂ N . Then notice thatthere is a corresponding decomposition M ⊗ d = ( N ′ ) ⊗ d ⊕ h N ⊗ M ⊗ d − i , which is a direct sum of S d -submodules. Taking S d -invariants on both sides resultsin the decomposition Γ d M = Γ d ( N ′ ) ⊕ h N ⊗ M ⊗ d − i S d (3.3)into k -submodules. The decomposition (3.3) then makes it possible to describe thekernel of the quotient map Γ d ( π ) : Γ d M ։ Γ d ( M/N )induced by projection π : M ։ M/N . More generally, we note the following.
Lemma 3.5.
Let A be a k -algebra. Suppose N ⊂ M is an inclusion of A -modulessuch that M = N ⊕ N ′ for some k -submodule N ′ ⊂ M . Then there is an exactsequence → h N ⊗ M ⊗ d − i S d −→ Γ d M Γ d ( π ) −−−−→ Γ d ( M/N ) → of Γ d A -module homomorphisms.Proof. It follows from (3.3) that the required exact sequence of Γ d A -modules isobtained by restriction from the exact sequence0 → h N ⊗ M ⊗ d − i −→ M ⊗ d π ⊗ d −−→ ( M/N ) ⊗ d → A ⊗ d -module homomorphisms. (cid:3) We introduce some additional notation. Suppose N , . . . , N r ⊂ M is a finitecollection of k -submodules of some M ∈ P k , and let µ ∈ Λ r ( d ). Then we write N ⊗ µ := N ⊗ µ ⊗ . . . ⊗ N ⊗ µ r r to denote the corresponding k -submodule of M ⊗ d and use the notation N µ := h N ⊗ µ i S d ⊂ Γ d M (3.4)for the k -submodule of S d -invariants.4. Wreath Products and Generalized Schur-Weyl Duality
Let us briefly recall the generalized Schur-Weyl duality [4] which establishes arelationship between a wreath product algebra A ≀ S d and a corresponding A -Schuralgebra via their respective actions on a common tensor space.4.1. Wreath products.
Fix a k -algebra A . The wreath product algebra A ≀ S d isthe k -module A ⊗ d ⊗ k S d , with multiplication defined by( x ⊗ ρ ) · ( y ⊗ σ ) := x ( yρ − ) ⊗ ρσ (4.1)for all x, y ∈ A ⊗ d and ρ, σ ∈ S d . If G is a finite group, then note for examplethat ( k G ) ≀ S d is isomorphic to the group algebra of the classical wreath product, G ≀ S d := G d ⋊ S d .Assume for the rest of the section that A is free as a k -module. We then identifythe tensor power A ⊗ d and group algebra k S d as subalgebras of A ≀ S d by setting A ⊗ d = A ⊗ d ⊗ S d , k S d = 1 A ⊗ d ⊗ k S d respectively.4.2. Generalized Schur-Weyl duality.
Suppose n, d ∈ N . Write V n := k n todenote the standard left M n ( k )-module, with basis elements v i := (0 , . . . , , . . . , i ∈ [1 , n ], considered as column vectors. Then for simplicity, let us writeV := V n ( A ) = k n ⊗ A to denote the corresponding left M n ( A )-module.We may identify V and A n as right A -modules, and it follows that the tensorspace , V ⊗ d , is naturally a right A ⊗ d -module. A right action of A ≀ S d on V ⊗ d isthen defined by setting w ( x · σ ) := ( wx ) σ, for w ∈ V ⊗ d , x ∈ A ⊗ d , and σ ∈ S d . (4.2) ELLULARITY OF GENERALIZED SCHUR ALGEBRAS 11
More explicitly, suppose w = w ⊗ · · · ⊗ w d and x = x ⊗ · · · ⊗ x d , for some w i ∈ Vand x i ∈ A . Then notice that( wx ) σ = ( w σ x σ ) ⊗ · · · ⊗ ( w dσ x dσ ) = ( wσ )( xσ )for any σ ∈ S d . Hence, by (4.1) we have w ( σ · x ) = w (( xσ − ) · σ ) = ( w ( xσ − )) σ = ( wσ ) x. It follows that (4.2) is well-defined.
Lemma 4.1 ([4, Lemma 5.7]) . The embedding S A ( n, d ) ֒ → M n ( A ) ⊗ d ∼ = End A ⊗ d (V ⊗ d ) defines an algebra isomorphism S A ( n, d ) ∼ = End A ≀ S d (V ⊗ d ) for all n, d ∈ N . Given n ≥ d , let ω ∈ Λ d ( n ) denote the weight ω = (1 d ) = (1 , . . . , , , . . . , n ( A )-module, notice that V ⊗ d is equal to theleft S A ( n, d )-module Γ ω V.For each weight µ ∈ Λ d ( n ), define a corresponding element v ⊗ µ := v ⊗ µ ⊗ . . . ⊗ v ⊗ µ n n in the tensor space V ⊗ d .The next result summarizes (5.15) and (5.17) of [4]. Proposition 4.2 ([4]) . Assume that n ≥ d . (i) There is a unique ( S A ( n, d ) , A ≀ S d ) -bimodule isomorphism S A ( n, d ) ξ ω ∼ −→ V ⊗ d which maps ξ ω v ⊗ ω . (ii) There is an algebra isomorphism, A ≀ S d ∼ −→ ξ ω S A ( n, d ) ξ ω , given by: ( x ⊗ . . . ⊗ x d ) ⊗ σ ξ x , σ ∗ · · · ∗ ξ x d d,dσ . (iii) End S A ( n,d ) (V ⊗ d ) ∼ = A ≀ S d . Cauchy Decompositions
The Cauchy decomposition for symmetric algebras via Schur modules [1] is ananalogue of Cauchy’s formula for symmetric functions [2, 18]. A correspondingdecomposition for divided powers [12, 17] is defined in terms of Weyl (or co-Schur)modules. In this section, we describe a generalized Cauchy decomposition (Theorem5.14) for divided powers of an (
A, B )-bimodule with respect to a given filtration onthe bimodule.5.1.
Weyl modules.
Weyl modules are defined in [1, Definition II.1.4] as the imageof a single map from a tensor product of divided powers of a module into a tensorproduct of exterior powers. We use an equivalent definition from the proof of [1,Theorem II.3.16]) which involves quotients of divided powers.Throughout the section, we fix some d ∈ N . Suppose λ ∈ Λ d ( N ), and let M ∈ P k .For each pair ( i, t ) with 1 ≤ i < l ( λ ) and 1 ≤ t ≤ λ i +1 , let us write λ ( i, t ) = ( λ , . . . , λ i − , λ i + t, λ i +1 − t, λ i +1 , . . . , λ m ) ∈ Λ d ( N ) . (5.1)Then write γ λ ( i,t ) : Γ λ ( i,t ) M → Γ λ M to denote the standard homomorphism corre-sponding to the matrix γ λ ( i,t ) := diag( λ , λ , . . . ) + tE i +1 ,i − tE i +1 ,i . Similarly, let γ tr λ ( i,t ) : Γ λ M → Γ λ ( i,t ) M denote the map corresponding to the trans-pose of the above matrix. Definition 5.1 ([1]) . Suppose M ∈ P k and λ ∈ Λ + d ( N ). Let (cid:3) λ ( M ) denote the k -submodule of Γ λ M defined by (cid:3) λ ( M ) := X i ≥ λ t +1 X t =1 Im( γ λ ( i,t ) ) ⊂ Γ λ M. The
Weyl module , W λ ( M ), is defined as the quotient k -module W λ ( M ) := Γ λ M (cid:14) (cid:3) λ ( M ) . Let A be a k -algebra and suppose now that M ∈ A -mod. Then (cid:3) λ ( M ) is a Γ d A -submodule of Γ λ M , since the standard homomorphisms are Γ d A -module maps. Itfollows that W λ ( M ) is a Γ d A -module. In particular, W λ ( k n ) is an S ( n, d )-module.5.2. The standard basis.
Consider a fixed partition λ = ( λ , λ , . . . ) ∈ Λ + d ( N ).The Young diagram of λ is the following subset of N × N :[ λ ] := { ( i, j ) | ≤ i ≤ l ( λ ) , ≤ j ≤ λ i } . Suppose B is a finite totally ordered set. Let Tab λ ( B ) denote the set of all functions T : [ λ ] → B , called tableaux ( of shape λ ).A tableau T will be identified with the diagram obtained by placing each valueT i,j := T ( i, j ) in the ( i, j )-th entry of [ λ ]. For example if T ∈ Tab (3 , ( B ), then wewrite T = T , T , T , T , T , (5.2)We say that a tableau T is row ( column ) standard if each row (column) is a nonde-creasing (increasing) function of i (resp. j ), and T is standard if it is both row andcolumn standard.Let St λ ( B ) ⊂ Tab λ ( B ) denote the subset of all standard tableaux. This subsetis nonempty if and only if l ( λ ) ≤ ♯ B . In particular, suppose l ( λ ) ≤ ♯ B and assumethe elements of B are listed as in (2.1). Then we write T λ = T λ ( B ) to denote thestandard tableau in St λ ( B ) with entries T λi,j := b B i for all ( i, j ) ∈ [ λ ]. For example,if d = 7, λ = (4 , ,
1) and B = [1 , T λ = 1 1 1 12 23 (5.3)Fix a free k -module V with finite ordered basis { x b } b ∈ B . If T ∈ Tab λ ( B ), thenfor q = l ( λ ) and i ∈ [1 , q ] we write T i := T ( i, − ) ∈ seq λ i ( B )to denote the to the i -th row of T , and we set x T := x T ⊗ · · · ⊗ x T q ∈ Γ λ V. Notice that the set of x T paramaterized by all row standard T ∈ Tab λ ( B ) forms abasis of Γ λ V .The following result describes a basis for Weyl modules. Proposition 5.2 ([1], Theorem III.3.16) . Let λ ∈ Λ + ( N ) and suppose V is a free k -module with a finite ordered basis { x b } b ∈ B . Then the Weyl module W λ ( V ) is alsoa free k -module, with basis given by the set of images { ¯ x T := π ( x T ) | T ∈ St λ ( B ) } under the canonical projection π : Γ λ V ։ Γ λ V (cid:14) (cid:3) λ ( V ) . ELLULARITY OF GENERALIZED SCHUR ALGEBRAS 13
This result shows for example that the Weyl module W λ ( V ) is nonzero if andonly if l ( λ ) ≤ ♯ B . Another consequence of the proposition is that W λ ( M ) is aprojective k -module for any M ∈ P k (cf. [17, p. 1013]).5.3. The Cauchy decomposition.
Suppose
M, N ∈ P k . The maps ψ d appearingin (2.11) can be generalized as follows. If λ ∈ Λ + d ( N ), let ψ λ ( M, N ) : Γ λ M ⊗ Γ λ N → Γ d ( M ⊗ N )denote the map defined via the compositionΓ λ M ⊗ Γ λ N ∼ −→ (Γ λ M ⊗ Γ λ N ) ⊗ . . . ⊗ (Γ λ m M ⊗ Γ λ m N ) ψ ⊗ ... ⊗ ψ −−−−−−→ Γ λ ( M ⊗ N ) ⊗ . . . ⊗ Γ λ m ( M ⊗ N ) ∇ −→ Γ d ( M ⊗ N ) , where the first map permutes tensor factors and the last map is multiplication inthe bialgebra Γ( M ⊗ N ).Let us write Γ λ : P k → P k to denote the tensor product of functorsΓ λ := Γ λ ⊗ · · · ⊗ Γ λ m defined in the same way as (2.10). Then it follows from Lemma 2.5 that the maps ψ λ ( M, N ) induce a natural transformation ψ λ : Γ λ ⊠ Γ λ → Γ λ ( − ⊗ − ) (5.4)of bifunctors P k × P k → P k .The following lemma is a special case of [12, Proposition III.2.6] which describesthe relationship between ψ -maps and standard homomorphisms. Lemma 5.3 ([12]) . Suppose λ ∈ Λ + d ( N ) , and set q = l ( λ ) . Given a pair U, V offree k -modules of finite rank, the following diagram is commutative Γ λ ( i,t ) U ⊗ Γ λ V Γ λ ( i,t ) U ⊗ Γ λ ( i,t ) V Γ λ U ⊗ Γ λ ( i,t ) V Γ λ U ⊗ Γ λ V Γ d ( U ⊗ V ) Γ λ U ⊗ Γ λ V id ⊗ γ tr λ ( i,t ) γ λ ( i,t ) ⊗ id ψ λ ( i,t ) γ tr λ ( i,t ) ⊗ id id ⊗ γ λ ( i,t ) ψ λ ψ λ for any i ∈ [1 , q − and t ∈ [1 , λ i +1 ] . Recalling the total order (cid:22) on Λ + d ( N ) from Definition 2.2, write λ + to denotethe immediate successor of a partition λ and set ( d ) + := ∞ . The Cauchy filtration is then defined as the chain0 = F ∞ ⊂ F ( d ) ⊂ · · · ⊂ F (1 ,..., = Γ d ( M ⊗ N )where F λ := P µ (cid:23) λ Im( ψ λ ).The following result describes the factors of this filtration. Theorem 5.4 ([12, Theorem III.2.7]) . Let
U, V be free k -modules of finite rank.Then for each λ ∈ Λ + d ( N ) , the map ψ λ induces an isomorphism ¯ ψ λ : W λ ( U ) ⊗ W λ ( V ) ∼ −→ F λ / F λ + which makes the following diagram commutative: Γ λ U ⊗ Γ λ V F λ W λ ( U ) ⊗ W λ ( V ) F λ / F λ + ψ λ ¯ ψ λ Hence, the associated graded module of the Cauchy filtration is M λ ∈ Λ + d ( N ) W λ ( U ) ⊗ W λ ( V ) . Proof.
We recall the proof from [12]. It follows by definition that W λ ( U ) ⊗ W λ ( V )is the quotient of Γ λ U ⊗ Γ λ V by the submodule (cid:3) λ ( U ) ⊗ Γ λ V + Γ λ U ⊗ (cid:3) λ ( V ).Hence, by Lemma 5.3 we have (cid:3) λ ( U ) ⊗ Γ λ V + Γ λ U ⊗ (cid:3) λ ( V ) ⊂ Im( ψ λ ( i,t ) ) ⊂ F λ + , since λ ( i, t ) > λ . This proves the existence of the induced map ¯ ψ λ satisfying thegiven commutative square. It is clear that ¯ ψ λ is surjective. Comparing the ranks ofΓ d ( U ⊗ V ) and L λ ∈ Λ + d ( N ) W λ ( U ) ⊗ W λ ( V ) shows that ¯ ψ λ must be an isomorphismfor each λ . (cid:3) Given free k -modules U, V ∈ P k with finite ordered bases { x b } b ∈ B and { y c } c ∈ C ,respectively, let F ′ λ ⊂ Γ d ( U ⊗ V ) denote the k -submodule generated by { ψ λ ( x S ⊗ y T ) | S ∈ St λ ( B ) , T ∈ St λ ( C ) } where F ′ λ is nonzero only if l ( λ ) ≤ min( ♯ B , ♯ C ). Corollary 5.5.
For each λ ∈ Λ + d ( N ) , the k -submodule F ′ λ ⊂ Γ d ( U ⊗ V ) is free, andthere is a corresponding decomposition: Γ d ( U ⊗ V ) = M λ F ′ λ , such that F λ = M µ ≥ λ F ′ µ for all λ ∈ Λ + d ( N ) . Proof.
Suppose λ ∈ Λ + d ( N ), and set T = St λ ( B ) × St λ ( C ). By Proposition 5.2, { ¯ x S ⊗ ¯ y T | ( S, T ) ∈ T } forms a basis of W λ ( U ) ⊗ W λ ( V ). So { ¯ ψ λ ( x S ⊗ y T ) | ( S, T ) ∈ T } gives a basis for F λ / F λ + by Theorem 5.4. This shows that the subset { ψ λ ( x S ⊗ y T ) | ( S, T ) ∈ T } ⊂ Γ d ( U ⊗ V )is linearly independent. Thus F ′ λ is a free k -submodule. It is also clear that F λ = F λ + ⊕ F ′ λ , and the required decompositions follow by induction. (cid:3) Bimodule filtrations.
In the remainder of this section, we fix a set { J ′ , . . . , J ′ r } of nonzero free k -submodules, J ′ i ⊂ J , such that setting J j := M ≤ i ≤ j J ′ i for j ∈ [1 , r ] (5.5)yields a chain 0 = J ⊂ J ⊂ · · · ⊂ J r = J of ( A, B )-bimodules.Recalling the notation (3.4), we then have for each µ ∈ Λ d ( r ) the following k -submodules of Γ d J : J ′ µ = h J ′⊗ µ i S d , J µ = h J ⊗ µ i S d . Note first that J µ is a Γ d ( A ⊗ B )-submodule of Γ d J , and hence a (Γ d A, Γ d B )-bimodule. It is also not difficult to check that there is a decomposition of J ⊗ d intofree k -submodules J ⊗ d = M µ ∈ Λ d ( r ) h J ′⊗ µ i . ELLULARITY OF GENERALIZED SCHUR ALGEBRAS 15
By taking S d -invariants on both sides, we thus obtain the following decompositionΓ d J = M µ ∈ Λ d ( r ) h J ′⊗ µ i ∩ Γ d J = M µ ∈ Λ d ( r ) J ′ µ . (5.6)Next recall that the dominance order on Λ d ( r ) is the partial order defined bysetting µ E ν if X i ≤ j µ i ≤ X i ≤ j ν i for j ∈ [1 , r ] . Notice that J µ ⊂ J ν if and only if µ D ν . We further have J ν = J ′ ν ⊕ P µ⊲ν J µ , andit follows by induction that J ν = M µ D ν J ′ µ (5.7)for all ν ∈ Λ d ( r ), which generalizes the decomposition (5.6) of Γ d J .Consider the map ∇ : Γ µ J → Γ d J given by r -fold (outer) multiplication in Γ( J ),for some µ ∈ Λ d ( r ). Note that the restriction ∇ µ : Γ ( µ ) ( J , . . . , J r ) → Γ d J (cid:16) resp. ′ ∇ µ : Γ ( µ ) ( J ′ , . . . , J ′ r ) → Γ d J (cid:17) is a (Γ d A, Γ d B )-bimodule (resp. k -module) homomorphism. Lemma 5.6.
Suppose ν ∈ Λ d ( r ) . Then(1) J ′ ν = Im ′ ∇ ν ,(2) ′ ∇ ν : Γ ( ν ) ( J ′ , . . . , J ′ r ) ∼ −→ J ′ ν is an isomorphism of k -modules,(3) J ν = P µ D ν Im ∇ µ , summing over µ ∈ Λ d ( r ) .Proof. For each µ ∈ Λ d ( r ), write M µ , M ′ µ to denote the images of Γ ( µ ) ( J , . . . , J r )and Γ ( µ ) ( J ′ , . . . , J ′ r ), respectively, under the map ∇ µ : Γ µ J → Γ d J . It is then clearfrom the definitions that M ′ µ ⊂ J ′ µ and similarly M µ = Im ∇ µ ⊂ J µ , for all µ .It follows inductively from the isomorphism (2.7) that there is a decompositionΓ d J = Γ d ( J ′ ⊕ · · · ⊕ J ′ r ) = M µ ∈ Λ d ( r ) M ′ µ ⊂ M µ ∈ Λ d ( r ) J ′ µ It thus follows from (5.6) that J ′ µ = M ′ µ ∼ = Γ ( µ ) ( J ′ , . . . , J ′ r ) which shows (1) and(2). Since J µ ⊂ J ν whenever µ D ν , it follows from (5.7) that J ν = M µ D ν M ′ µ ⊂ X µ D ν M µ ⊂ X µ D ν J µ ⊂ J ν showing (3). (cid:3) Recall the lexicographic ordering ≤ on Λ d ( r ) from Definition 2.2, and notice thatthere is a chain of (Γ d A, Γ d B )-sub-bimodules0 ⊂ Γ d ( J ) = J ≥ ( d, ,..., ⊂ · · · ⊂ J ≥ ν ⊂ · · · ⊂ J ≥ (0 ,..., ,d ) = Γ d J where J ≥ ν := P µ ≥ ν J µ for each ν ∈ Λ d ( r ). Since the lexicographic ordering refinesthe dominance order, it follows from (5.7) that J ≥ ν = J >ν ⊕ J ′ ν (5.8)for all ν . Thus J ≥ ν = X µ ≥ ν Im( ∇ µ )by the preceding lemma. This allows us to describe the quotients J ≥ ν /J >ν asfollows. Proposition 5.7.
Let ν ∈ Λ d ( r ) . Then ∇ ν induces an isomorphism ¯ ∇ ν : Γ ( ν ) ( J /J , . . . , J r /J r − ) ∼ = J ≥ ν /J >ν which yields a commutative square of (Γ d A, Γ d B ) -bimodule homomorphisms Γ ( ν ) ( J , . . . , J r ) J ≥ ν Γ ( ν ) ( J /J , . . . , J r /J r –1 ) J ≥ ( ν ) /J > ( ν ) ∇ ν π ν π ¯ ∇ ν where π ν denotes the tensor product of functorial maps Γ ν j ( π j ) associated to theprojections, π j : J j → J j /J j − , for j = 1 , . . . , r , and where π is also projection.Proof. We first verify that ker π ν ⊂ J >ν in order to show the existence of themap ¯ ∇ ν satisfying the above diagram. If 1 ≤ j ≤ r , consider the (Γ d A, Γ d B )-sub-bimodule K j := Γ ( ν ) ( J , . . . , J r ) ⊗ ker Γ ν j ( π j ) ⊗ Γ ( ν ) ( J , . . . , J r )where ν = ( ν , . . . , ν j − , , . . . ,
0) and ν = (0 , . . . , , ν j +1 , . . . , ν r ). Then ker π ν = P rj =1 K j , and we must show that K j ⊂ J >ν for all j .Now K j = 0, if either j = 1 or ν j = 0. If K j = 0 and 1 ≤ t ≤ ν j , let ν ( j, t ) ∈ Λ d ( r ) be defined as in (5.1). Since ν ( j, > ν , it suffices to show that ∇ ν ( K j ) ⊂ Im ∇ ν ( j, for all such j . The fact that ν and ν ( j,
1) are equal except forentries in the j -th and ( j − r = 2.So we may assume ν = ( ν , ν ). Then for j = 2, we have ν (2 ,
1) = ( ν +1 , ν − K = ker( π ν ), and it follows by Lemma 3.5 that K = Γ ν J ⊗ J (1 ,ν − ⊂ Γ ( ν ) ( J , J ) . Notice by Lemma 5.6 that J (1 ,ν − is equal to the image of the map ∇ (1 ,ν − : J ⊗ Γ ν − ( J ) → Γ ν ( J ) . By associativity of multiplication in Γ( J ), we also have a commutative diagramΓ (( ν , ,ν –1)) ( J , J , J )Γ ( ν ) ( J , J ) Γ ( ν (2 , ( J , J ) J ≥ ν id ⊗∇ (1 ,ν − ∇⊗ id ∇ ν ∇ ν (2 , It follows that ∇ ν ( K ) ⊂ Im ∇ ν (2 , , which shows the existence of ¯ ∇ ν . To completethe proof, note that the restriction π ν | Γ ( ν ) ( J ′ ,...,J ′ r ) is a k -module isomorphism. Themap ( π ◦ ∇ ν ) | Γ ( ν ) ( J ′ ,...,J ′ r ) is also a k -module isomorphism by Lemma 5.6. It followsthat ¯ ∇ ν is an isomorphism by commutativity. (cid:3) Multitableaux.
Suppose { B j } j ∈ [1 ,r ] is a collection of finite totally orderedsets, and let λ ∈ Λ + d ( N ) r be an r -multipartition. Elements of the set Tab λ ( B , . . . , B r ) := Tab λ (1) ( B ) × · · · × Tab λ ( r ) ( B r ) . are called multitableaux of shape λ (or λ - multitableaux ).We say that a λ -multitableau, T = ( T (1) , . . . , T ( r ) ), is standard if each component T ( j ) is a standard λ ( j ) -tableau. The subset of standard λ -multitableaux is denoted St λ ( B ∗ ) = St λ ( B , . . . , B r ) . ELLULARITY OF GENERALIZED SCHUR ALGEBRAS 17
If ( n , . . . , n r ) ∈ N r is the sequence of integers with n j := ♯ B j for all j , then itfollows from (5.3) that St λ ( B ∗ ) is non-empty if and only if λ belongs to the subsetΛ + d ( n , . . . , n r ) ⊂ Λ + d ( N ) r . In this case, we write T λ = T λ ( B ∗ ) to denote thestandard λ -multitableau T λ := ( T λ (1) , . . . , T λ ( r ) ) . (5.9)Suppose ν = ( ν , . . . , ν r ) ∈ Λ d ( r ). There is a corresponding r -multipartition( ν ) := (( ν ) , ( ν ) , . . . , ( ν r )) ∈ Λ + d ( N ) r . For any m ∈ N , let us write (1 m ) :=(1 , . . . , ∈ Λ + m ( N ), and set (1 ) = 0. Then we also have an element( ν ) ′ := ((1 ν ) , (1 ν ) , . . . , (1 ν r )) ∈ Λ + ν ( N ) . Recalling the total order (cid:22) from Definition 2.4, notice that ( ν ) ′ (cid:22) λ (cid:22) ( ν ) for all λ ∈ Λ + ν ( N ). We also write λ + to denote the immediate successor of any λ ∈ Λ + d ( N ) r and set (( d )) + = ∞ .5.6. Generalized Weyl modules.
Given λ ∈ Λ + d ( N ) r and projective modules M j ∈ P k for j ∈ [1 , r ], we will use the notationΓ λ ( M ∗ ) := O j Γ λ ( j ) M j , W λ ( M ∗ ) := O j W λ ( j ) M j in what follows. The outer tensor product − ⊠ − , defined in Section 2.7, yieldscorresponding functors Γ λ , W λ : P × r k → P k defined byΓ λ := Γ λ (1) ⊠ · · · ⊠ Γ λ ( r ) and W λ := W λ (1) ⊠ · · · ⊠ W λ ( r ) . Since Weyl modules are quotients of divided powers, it follows that there is a naturalprojection π : Γ λ ։ W λ .Suppose V , . . . , V r ∈ P k are free k -modules, and suppose { x ( j ) b } b ∈ B j is a finiteordered basis of V j for each j ∈ [1 , r ]. Given a multitableau T ∈ Tab λ ( B , . . . , B r ),there is a corresponding element x T := O j x ( j ) T ( j ) ∈ Γ λ ( V ∗ )whose image in W λ ( V ∗ ) is denoted ¯ x T := π ( x T ). The next result follows easilyfrom Proposition 5.2. Lemma 5.8.
Let λ ∈ Λ + d ( N ) r be an r -multipartition, and let V , . . . , V r be free k -modules with bases as above. The set of images { ¯ x T | T ∈ St λ ( B , . . . , B r ) } formsa basis of the free k -module W λ ( V ∗ ) parametrized by standard λ -multitableaux. Inparticular, we have W λ ( V ∗ ) = 0 unless λ ∈ Λ + d ( ♯ B , . . . , ♯ B r ) . Suppose ν ∈ Λ d ( r ) and fix some projective modules M j , N j ∈ P k for j ∈ [1 , r ].Using notation similar to the above, we writeΓ ( ν ) ( M ∗ ⊗ N ∗ ) := O j Γ ν j ( M j ⊗ N j ) . Given λ ∈ Λ + ν ( N ), we then define a map ψ λ : Γ λ ( M ∗ ) ⊗ Γ λ ( N ∗ ) → Γ ( ν ) ( M ∗ ⊗ N ∗ )via the composition (cid:8) O j Γ λ ( j ) M j (cid:9) ⊗ (cid:8) O j Γ λ ( j ) N j (cid:9) (5.10) ∼ = O j (cid:8) Γ λ ( j ) ( M j ) ⊗ Γ λ ( j ) ( N j ) (cid:9) ψ ⊗ ... ⊗ ψ −−−−−−→ O j Γ ν j ( M j ⊗ N j ) . Note that if M j ∈ A -mod and N j ∈ B -mod for all j , then ψ λ is a homomorphismof (Γ d A, Γ d B )-bimodules by Lemma 2.5.1. Generalized Cauchy filtrations of bimodules.
Fix a chain ( J j ) j ∈ [0 ,r ] of( A, B )-bimodules. For each j ∈ [1 , r ], suppose there exists an isomorphism α j : J j /J j − ∼ −→ U j ⊗ V j (5.11)of ( A, B )-bimodules for some U j ∈ A -mod and V j ∈ B -mod. Assume for all j that U j and V j are free as k -modules, with finite ordered bases { x ( j ) b } b ∈ B j and { y ( j ) c } c ∈ C j ,respectively. Assume further that { J ′ j } j ∈ [ r ] is any collection of free k -submodulesof J r such that (5.5) holds.We first define a filtration of Γ ( ν ) ( U ∗ ⊗ V ∗ ) for some fixed weight ν ∈ Λ d ( r ). Foreach r -multipartition λ ∈ Λ + ν ( N ) r , let us write F λ , ( ν ) := X λ ≤ µ ≤ ( ν ) F µ (1) ( U , V ) ⊗ · · · ⊗ F µ ( r ) ( U r , V r )which is a sum of sub-bimodules of Γ ( ν ) ( U ∗ ⊗ V ∗ ). It follows that there is a chainof sub-bimodules:0 =: F ( ν ) + , ( ν ) ⊂ F ( ν ) , ( ν ) ⊂ · · · ⊂ F ( ν ) ′ , ( ν ) = Γ ( ν ) ( U ∗ ⊗ V ∗ ) . (5.12)Recalling (5.10), notice that for each λ ∈ Λ + ν ( N ) r we have F λ , ( ν ) = X λ (cid:22) µ (cid:22) ( ν ) Im( ψ µ ) . Note also that F λ , ( ν ) contains the k -submodule F ′ λ := O F ′ λ ( j ) ( U j , V j ) . It then follows by Corollary 5.5 that F ′ λ is a free k -submodule, with the set { ψ λ ( x S ⊗ y T ) | S ∈ St λ ( B , . . . , B r ) , T ∈ St λ ( C , . . . , C r ) } (5.13)as a basis. Proposition 5.9.
Suppose ν ∈ Λ d ( r ) . Then for each λ ∈ Λ + ν ( N ) , the map ψ λ : Γ λ ( U ∗ ) ⊗ Γ λ ( V ∗ ) → F λ , ( ν ) induces an isomorphism ¯ ψ λ : F λ , ( ν ) / F λ + , ( ν ) ∼ −→ W λ ( U ∗ ) ⊗ W λ ( V ∗ ) of bimodules. We also have decompositions Γ ( ν ) ( U ∗ ⊗ V ∗ ) = M λ ∈ Λ + ν ( N ) F ′ λ , F λ , ( ν ) = M λ (cid:22) µ (cid:22) ( ν ) F ′ µ (5.14) into free k -submodules. We now wish to lift the filtrations (5.12), for varying ν , to a single filtration ofΓ d J , with J = J r as above. First note that there is an isomorphism φ ν : J ≥ ν /J >ν ∼ −→ Γ ( ν ) ( U ∗ ⊗ V ∗ )satisfying the following commutative triangle of (Γ d A, Γ d B )-bimodule isomorphisms: N Γ ν j ( J j /J j − ) N Γ ν j (cid:0) U j ⊗ V j (cid:1) J ≥ ν /J >ν Γ ( ν ) ( α ∗ )¯ ∇ ν φ ν (5.15) ELLULARITY OF GENERALIZED SCHUR ALGEBRAS 19 where Γ ( ν ) ( α ∗ ) = N Γ ν j ( α j ) is a tensor product of isomorphisms induced by themaps (5.11) and ¯ ∇ ν is defined in Proposition 5.7. We then have a surjective mapˆ φ ν : J ≥ ν ։ Γ ( ν ) ( U ∗ ⊗ V ∗ )obtained by composing φ ν with the projection π : J ≥ ν ։ J ≥ ν /J >ν . Definition 5.10.
Suppose λ ∈ Λ + d ( N ) r and set ν = | λ | . Then define J λ to be thesub-bimodule of J ≥ ν , corresponding to the inverse image of F λ , ( ν ) under the map φ ν considered above. The generalized Cauchy filtration of Γ d J is then defined asthe chain 0 = J ∞ ⊂ J (( d )) ⊂ · · · ⊂ J λ + ⊂ J λ ⊂ · · · ⊂ J ((1 d )) = Γ d J (5.16)of (Γ d A, Γ d B )-bimodules parametrized by multipartitions λ ∈ Λ + d ( N ) r .We next define a decomposition of Γ d J via certain k -submodules, J ′ λ ⊂ J λ .Recall from (5.5) that J j = J ′ j ⊕ J j − , for all j . For each j ∈ [1 , r ], let α ′ j : J ′ j ∼ −→ U j ⊗ V j denote the isomorphism defined via the composition J ′ j J j J j /J j − U j ⊗ V j . α j Similar to (5.15), there is a resulting k -module isomorphism φ ′ ν : J ′ ν ∼ −→ Γ ( ν ) ( U ∗ ⊗ V ∗ )satisfying the following commutative triangle of isomorphisms:Γ ( ν ) ( J ′∗ ) Γ ( ν ) ( U ∗ ⊗ V ∗ ) J ′ ν Γ ( ν ) ( α ′∗ ) ′ ∇ ν φ ′ ν (5.17)where Γ ( ν ) ( α ′∗ ) := O Γ ν j ( α ′ j )and where ′ ∇ ν is restriction of r -fold multiplication as in Lemma 5.6.(i). We write J ′ λ := ( φ ′ ν ) − ( F ′ λ )to denote the inverse image of F ′ λ under φ ′ ν . Lemma 5.11.
There exist decompositions into free k -submodules Γ d J = M λ ∈ Λ + d ( N ) r J ′ λ , and J λ = M λ (cid:22) µ ≺∞ J ′ µ for each λ . Proof.
It follows by definition from (5.15) and (5.17) that φ ′ ν can be obtained fromˆ φ by restriction. In particular, we have a commutative diagram: J ′ ν Γ ( ν ) ( U ∗ ⊗ V ∗ ) J ≥ ν Γ ( ν ) ( U ∗ ⊗ V ∗ ) φ ′ ν ˆ φ ν (5.18)Since J ≥ ν = J >ν ⊕ J ′ ν by (5.8), we further have a decompositionˆ φ − ν ( N ) = J >ν ⊕ ( φ ′ ν ) − ( N ) (5.19) for any k -submodule N ⊂ Γ ( ν ) ( U ∗ ⊗ V ∗ ). If we set N = F λ , ( ν ) in the above, then itfollows from (5.14) that J λ = J >ν ⊕ M λ (cid:22) µ (cid:22) ( ν ) J ′ λ for each λ ∈ Λ ν ( N ). The decomposition of J λ now follows by induction since J >ν = J ( ν + ) , where ν + denotes an immediate successor of ν in the lexicographicorder on Λ d ( r ). The decomposition for Γ d J = J (1 d ) follows as a special case. (cid:3) Now suppose λ ∈ Λ + d ( N ) r . To each element of the basis (5.13), we associate acorresponding element in J ′ λ , defined by z S , T := (cid:0) ′ ∇ ν ◦ Γ ( ν ) ( α ′∗ ) − ◦ ψ λ (cid:1) ( x S ⊗ y T ) . (5.20)Since the map appearing in (5.20) is a composition of isomorphisms, it follows thatthe set { z S , T | S ∈ St λ ( B ∗ ) , T ∈ St λ ( C ∗ ) } forms a basis of J ′ λ .Let ( m , . . . , m r ) ∈ N r be the sequence defined by m j := min( ♯ B j , ♯ C j )for all j , and set Λ := Λ + r ( m , . . . , m r ) . Remark 5.12.
Suppose λ ∈ Λ + ( N ) r . If λ belongs to Λ ⊂ Λ + ( N ) r , then St ( B ∗ )and St ( C ∗ ) are both non-empty since they contain the elements T λ = T λ ( B ∗ ) and T λ = T λ ( C ∗ ) defined in (5.9), respectively. We thus have J ′ λ = 0 if and only if λ ∈ Λ .Let λ ∈ Λ . Since J λ = J ′ λ ⊕ J λ + by Lemma 5.11, it follows that J λ / J λ + is a free k -module with basis { ¯ z S , T | S ∈ St λ ( B ∗ ) , T ∈ St λ ( C ∗ ) } where ¯ x := x + J λ + denotes the image of x ∈ J λ in the quotient. Definition 5.13.
Given λ ∈ Λ , define a pair of k -submodules U λ , V λ ⊂ J λ / J λ + generated by the subsets (cid:8) ¯ z S , T λ | S ∈ St λ ( B ∗ ) (cid:9) and (cid:8) ¯ z T λ , T | T ∈ St λ ( C ∗ ) (cid:9) , respectively. It is then clear that U λ is a Γ d A -submodule of the (Γ d A, Γ d B )-bimodule J λ / J λ + , and V λ is a Γ d B -submodule.The following analogue of Theorem 5.4 is the main result in this section. Theorem 5.14 (Generalized Cauchy Decomposition) . Suppose λ ∈ Λ . Then themap of k -modules defined by α λ : J λ / J λ + → U λ ⊗ V λ : ¯ z S , T ¯ z S , T λ ⊗ ¯ z T λ , T , for all ( S , T ) ∈ St λ ( B ∗ ) × St λ ( C ∗ ) , is an isomorphism of (Γ d A, Γ d B ) -bimodules.The associated graded module of the generalized Cauchy filtration is thus given by M λ ∈ Λ U λ ⊗ V λ . ELLULARITY OF GENERALIZED SCHUR ALGEBRAS 21
Proof.
Write φ λ : J λ → F λ , ( ν ) to denote the map obtained from ˆ φ ν by restriction.There is an induced bimodule isomorphism¯ φ λ : J λ / J λ + ∼ −→ F λ , ( ν ) / F λ + , ( ν ) which follows from the definitions by using the decompositions J λ = J ′ λ ⊕ J λ + and F λ , ( ν ) = F ′ λ ⊕ F λ + , ( ν ) .Hence by Proposition 5.9, there is an isomorphism ϕ λ making the upper righttriangle commute in the following diagram J λ / J λ + F λ , ( ν ) / F λ + , ( ν ) U λ ⊗ V λ W λ ( U ∗ ) ⊗ W λ ( V ∗ ) . α λ ¯ φ λ ¯ ψ λ ϕ λ ϕ ′ λ ⊗ ϕ ′′ λ (5.21)In the bottom arrow, the map ϕ ′ λ (resp. ϕ ′′ λ ) denotes the homomorphism obtainedby composing ϕ λ with the embedding W λ ( U ∗ ) ∼ −→ W λ ( U ∗ ) ⊗ ¯ y T λ (resp. W λ ( V ∗ ) ∼ −→ ¯ x T λ ⊗ W λ ( V ∗ )) . In order to complete the proof, it suffices to show that the lower triangle in (5.21)is a commutative triangle of isomorphisms. For this, we compute: ϕ λ (¯ x S ⊗ ¯ y T ) = ( ¯ φ − λ ◦ ¯ ψ λ )(¯ x S ⊗ ¯ y T )= ¯ φ − λ ( ψ λ ( x S ⊗ y T ) ) by Prop. 5.9= ( φ ′ λ ) − ◦ ψ λ ( x S ⊗ y T ) by (5.18) and (5.19)= ¯ z S , T . It follows that ϕ ′ λ ⊗ ϕ ′′ λ is an isomorphism since ϕ ′ λ ⊗ ϕ ′′ λ (¯ x S ⊗ ¯ y T ) = ¯ z S , T λ ⊗ ¯ z T λ , T for all ( S , T ) ∈ St λ ( B ∗ ) × St λ ( C ∗ ). Since it is now clear that the lower triangle iscommutative, the proof is complete. (cid:3) It follows from the proof of the theorem that U λ and V λ are each isomorphic toa respective (generalized) Weyl module. In the case B = A op , we call U λ (resp. V λ )a left (resp. right) Weyl submodule of the Γ d A -bimodule J λ / J λ + .6. Cellular Algebras
Assume throughout this section that k is a noetherian integral domain. We firstrecall the definition of cellular algebras from [8], along with the reformulation givenin [16]. We then use the generalized Cauchy decomposition to describe a cellularstructure on generalized Schur algebras S A ( n, d ).6.1. Definition of cellular algebras.Definition 6.1 (Graham-Lehrer) . An associative k –algebra A is called a cellularalgebra with cell datum ( I, M, C, τ ) if the following conditions are satisfied:(C1) ( I, D ) is a finite partially ordered set. Associated to each λ ∈ I is a finiteset M ( λ ). The algebra A has a k -basis C λS,T , where ( S, T ) runs through allelements of M ( λ ) × M ( λ ) for all λ ∈ I .(C2) The map τ is an anti-involution of A such that τ ( C λS,T ) = C λT,S . (C3) For each λ ∈ I and S, T ∈ M ( λ ) and each a ∈ A , the product aC λS,T can bewritten as ( P U ∈ M ( λ ) r a ( U, S ) C λU,T ) + r ′ , where r ′ is a linear combinationof basis elements with upper index µ strictly larger than λ , and where thecoefficients r a ( U, S ) ∈ k do not depend on T .Let A be a cellular algebra with cell datum ( I, M, C, τ ). Given λ ∈ I , it is clearthat the set J ( λ ) spanned by the C µS,T with µ D λ is a τ –invariant two sided idealof A (see [8]). Let J ( ⊲λ ) denote the sum of ideals J ( µ ) with µ ⊲ λ .For λ ∈ I , the standard module ∆( λ ) is defined as follows: as a k -module, ∆( λ )is free with basis indexed by M ( λ ), say { C λS | S ∈ M ( λ ) } ; for each a ∈ A , the actionof a on ∆( λ ) is defined by aC λS = P U r a ( U, S ) C λU where the elements r a ( U, S ) ∈ k are the coefficients in (C3). Any left A -module isomorphic to ∆( λ ) for some λ willalso be called a standard module. Note that for any T ∈ M ( λ ), the assignment C λS C λS,T + J ( ⊲λ ) defines an injective A –module homomorphism from ∆( λ ) to J ( λ ) /J ( ⊲λ ).6.2. Basis-free definition of cellular algebras.
In [16], K¨onig and Xi providean equivalent definition of cellular algebras which does not require specifying aparticular basis. This definition can be formulated as follows.
Definition 6.2 (K¨onig-Xi) . Suppose A is a k -algebra with an anti-involution τ .Then a two-sided ideal J in A is called a cell ideal if, and only if, J = τ ( J ) andthere exists a left ideal ∆ ⊂ J such that ∆ is finitely generated and free over k andsuch that there is an isomorphism of A -bimodules α : J ∼ −→ ∆ ⊗ τ (∆) making thefollowing diagram commutative: J ∆ ⊗ τ (∆) J ∆ ⊗ τ (∆) ατ x ⊗ y τ ( y ) ⊗ τ ( x ) α We say that a decomposition A = J ′ ⊕ · · · ⊕ J ′ r (for some r ) into k -submoduleswith τ ( J ′ j ) = J ′ j for each j = 1 , . . . , r is a cellular decomposition of A if setting J j := L ≤ i ≤ j J ′ i gives a chain of ( τ -invariant) two-sided ideals0 = J ⊂ J ⊂ J ⊂ · · · ⊂ J r = A such that the quotient J j /J j − is a cell ideal (with respect to the anti-involutioninduced by τ on the quotient) of A/J j − .The above chain of ideals in A is called a cell chain . For each ideal J j in a cellchain, we write ∆ j ⊂ J j /J j − , α j : J j /J j − ∼ −→ ∆ j ⊗ τ (∆ j ) (6.1)to denote the corresponding left ideal and A -bimodule isomorphism. Since J j = J ′ j ⊕ J j − for all j , we have a k -module isomorphism α ′ j : J ′ j ∼ = ∆ j ⊗ τ (∆ j ) definedas the composition α ′ j : J ′ j J j J j /J j − ∆ j ⊗ τ (∆ j ) . α j It then follows by definition that we have a commutative diagram J ′ j ∆ j ⊗ k τ (∆ j ) J ′ j ∆ j ⊗ k τ (∆ j ) α ′ j i x ⊗ y τ ( y ) ⊗ τ ( x ) α ′ j (6.2)of k -module isomorphisms. ELLULARITY OF GENERALIZED SCHUR ALGEBRAS 23
Lemma 6.3 (K¨onig-Xi, [16]) . Let A be an associative k -algebra with an anti-involution τ . Then A is a cellular algebra in the sense of [8] if and only if A has acellular decomposition.Proof. We summarize the proof from [16]. Let A be a cellular algebra with celldatum ( I, M, C, τ ). First, suppose λ ∈ I is maximal. Then J = J ( λ ) is a two-sidedideal by (C3) and J = τ ( J ) by (C2). Fix any element T λ ∈ M ( λ ). Define ∆ as the k –span of C λS,T λ where S varies. Defining α by sending C λS,T λ ⊗ τ ( C λT,T λ ) to C λS,T gives the required isomorphism. Thus J ( λ ) is a cell ideal.Next, choose any enumeration λ , . . . , λ r of the elements of I such that i < j whenever λ j ⊲ λ i . Set J ′ j ⊂ A (for each j ) equal to the k –span of all C λ j S,T (forvarying
S, T ). We have τ ( J ′ j ) = J ′ j by (C2). Since J ( λ j ) = J ′ j L J ( ⊲λ j ) for all j , itfollows that A = L j J ′ j is a cellular decomposition.For the converse, consider the index set I = { , . . . , r } with the reversed ordering1 ⊲ · · · ⊲ r . Choose a k -basis { x ( j ) b } b ∈ B j of ∆ j , for each j ∈ I . Setting C jb,c ∈ J ′ j to be the inverse image of x ( j ) b ⊗ τ ( x ( j ) c ) (for b, c ∈ B j ) under α ′ j (for j ∈ I ) givesa k -basis for A of the form (C1). Since ∆ j is a left A -module, (C3) is satisfied.Finally, (C2) follows from the required commutative diagram and the τ -invarianceof J ′ j . It follows that { C jb,c } is a cellular basis. (cid:3) From now on, we say that an algebra A with anti-involution τ is cellular if eitherof the equivalent statements in Lemma 6.3 is satisfied. The proof of the lemmashows that each ideal ∆ j (for j = 1 , . . . , r ) for a cellular algebra A is a standardmodule.6.3. Matrix algebras.
Consider the matrix ring, M n ( k ), with matrix transpose, tr , as anti-involution. Let us write, c : V n ⊗ V tr n ∼ −→ M n ( k ), to denote the isomor-phism mapping v i ⊗ v tr j E ij for all i, j ∈ [1 , n ].Now suppose A is an algebra with anti-involution τ , and let J be a cell idealwith defining isomorphism α : J ∼ −→ ∆ ⊗ τ (∆). ThenM n ( J ) := M n ( k ) ⊗ J is a cell ideal of the matrix ring M n ( A ) with respect to the anti-involution tr ⊗ τ .The corresponding isomorphism is the map c − ( α ) : M n ( J ) ∼ −→ V n (∆) ⊗ V tr n ( τ (∆))defined by the compositionM n ( k ) ⊗ J c − ⊗ α −−−−−−→ (cid:0) V n ⊗ V tr n (cid:1) ⊗ (∆ ⊗ τ (∆)) ∼ −−→ V n ⊗ ∆ ⊗ V tr n ⊗ τ (∆) . More generally, we have the following.
Lemma 6.4.
Suppose A is a cellular algebra with anti-involution τ and cell chain ( J j ) j ∈ [1 ,r ] . Then the matrix ring M n ( A ) is cellular with anti-involution tr ⊗ τ andcell chain (M n ( J j )) j ∈ [1 ,r ] , where M n ( J j ) := M n ( k ) ⊗ J j for all j .Proof. It follows from the preceding paragraph that the ideals, M n ( J j ), form a cellchain, since M n ( J j ) / M n ( J j − ) ≃ M n ( k ) ⊗ ( J j /J j − ) as M n ( A )-bimodules. It isalso clear that M n ( A ) has a cellular decompositionM n ( A ) = M M n ( J ′ j )where A = L J ′ j denotes a corresponding cellular decomposition of A . (cid:3) Cellularity of generalized Schur algebras.
We now describe a cellularstructure for generalized Schur algebras S A ( n, d ). In this case, the generalizedCauchy filtration forms a cell chain, with the Weyl submodules from Theorem 5.14as standard modules. Theorem 6.5.
Suppose A is a cellular algebra with anti-involution τ . Then thegeneralized Schur algebra S A ( n, d ) is a cellular algebra, with respect to the anti-involution τ := ( tr ⊗ τ ) ⊗ d , for all n, d ∈ N .Proof. If A is cellular then so is M n ( A ), by Lemma 6.4. Since S A ( n, d ) = Γ d M n ( A ),it suffices to show that Γ d A is cellular, with respect to the anti-involution τ = τ ⊗ d .Suppose that A = J ′ ⊕ · · · ⊕ J ′ r is a cellular decomposition of A , with corre-sponding cell chain 0 = J ⊂ J ⊂ · · · ⊂ J r = A. For each j ∈ [1 , r ], suppose { x ( j ) b } b ∈ B j and { y ( j ) b } b ∈ B j are k -bases of ∆ j and τ (∆ j ),respectively, such that y ( j ) b := τ ( x ( j ) b ) for all j , and let ∆ j and α j be as in (6.1).Considering Λ = Λ + ( ♯ B , . . . , ♯ B r ) as a totally ordered subset of Λ + d ( N ) r byrestricting the order (cid:22) in Definition 2.4, it follows from Lemma 5.11 and Remark5.12 that we have decompositionsΓ d J = M λ ∈ Λ J ′ λ , and J λ = M µ (cid:23) λ J ′ µ for each λ ∈ Λ , (6.3)since J ′ λ = 0 if λ / ∈ Λ .Notice that τ = τ ⊗ d coincides with the map Γ d ( τ ) : Γ d A → Γ d A induced by thefunctor Γ d . To complete the proof, we need to show that the left-hand side of (6.3)gives a cellular decomposition of Γ d A with respect to this anti-involution.Let ∆ λ be the the left Weyl submodule U λ ⊂ J λ / J λ + of Theorem 5.14. Then itremains to check the following hold for each λ ∈ Λ :(i) τ ( J ′ λ ) = J ′ λ ,(ii) τ (∆ λ ) = V λ ,(iii) J λ / J λ + is a cell ideal.Assuming (i) and (ii) hold for each λ , (iii) will follow from the commutativity ofthe diagram J λ / J λ + ∆ λ ⊗ τ (∆ λ ) J λ / J λ + ∆ λ ⊗ τ (∆ λ ) α λ τ x ⊗ y τ ( y ) ⊗ τ ( x ) α λ where α λ is the Γ d A -bimodule isomorphism from Theorem 5.14.Now fix λ ∈ Λ , and set ν = | λ | . Then J ′ λ , ∆ λ , and J λ / J λ + have k -bases givenby the sets { z S , T | S , T ∈ St ( B ∗ ) } , { ¯ z S , T λ | S ∈ St ( B ∗ ) } , { ¯ z S , T | S , T ∈ St ( B ∗ ) } respectively, where z S , T ∈ J ′ λ is defined in (5.20). It follows that each of theconditions (i)-(iii) will be satisfied provided that τ ( z S , T ) = z T , S for all S , T ∈ St ( B ∗ ). ELLULARITY OF GENERALIZED SCHUR ALGEBRAS 25
We claim that the following diagram is commutative:Γ λ (∆ ∗ ) ⊗ Γ λ ( τ (∆ ∗ )) Γ ( ν ) (∆ ∗ ⊗ τ (∆ ∗ )) Γ ( ν ) ( J ′∗ ) J ′ ν Γ λ (∆ ∗ ) ⊗ Γ λ ( τ (∆ ∗ )) Γ ( ν ) (∆ ∗ ⊗ τ (∆ ∗ )) Γ ( ν ) ( J ′∗ ) J ′ ν , ψ λ tw ◦ ( Γ λ ( τ ) ⊗ Γ λ ( τ ) ) Γ ( ν ) ( tw ◦ ( τ ⊗ τ ))Γ ( ν ) ( α ′∗ ) ∇ ν Γ ( ν ) ( τ ) τψ λ Γ ( ν ) ( α ′∗ ) ∇ ν with the first (middle) vertical map(s) induced by the action of Γ λ (resp. Γ ( ν ) ) con-sidered as a functor P × r k → P k . The commutativity of the left-hand square can bechecked using the definition of ψ λ together with Lemma 2.5. The commutativity ofthe middle square follows from the functoriality of Γ ( ν ) and diagram (6.2). Finally,the commutativity of the right-hand square follows by Lemma 3.4. We thus have τ ( z S , T ) = z T , S for all S , T ∈ St ( B ∗ ), and the proof is complete. (cid:3) Let us write Λ op to denote the set Λ with opposite total ordering. Then itfollows from the above proofs of Lemma 6.3 and Theorem 6.5 that the set (cid:8) z S , T | λ ∈ Λ op , S , T ∈ St λ ( B , . . . , B r ) (cid:9) . is a cellular basis for Γ d A . A corresponding cellular basis for S A ( n, d ) can beobtained in a similar way, by replacing A by M n ( A ).In the next example, we describe an explicit cellular basis for a special case ofa generalized Schur algebra of the form S Z ( n, d ), where Z is a zig-zag algebra.We essentially follow the definition in [15], using slightly different notation. Notealso that we only consider Z as an ordinary non-graded algebra, rather than a Z / Example 6.6 (Zig-zag algebra) . We consider the zig-zag algebra associated to thequiver below. Q : • • • a a a a Recall from [15, Section 7.9] that the extended zig-zag algebra , ˜ Z , is defined in thiscase as the quotient of the path algebra k Q modulo the following relations:(1) All paths of length three or greater are zero.(2) All paths of length two that are not cycles are zero.(3) All length-two cycles based at the same vertex are equivalent.(4) a a = 0.The length zero paths are denoted e , e , e and correspond to standard idempo-tents, with e i a ij e j = a ij for all admissible i, j . Let e := e + e ∈ ˜ Z . Then thecorresponding zig-zag algebra is Z := e ˜ Ze ⊂ ˜ Z . Then Z is a cellular algebra, withanti-involution defined by τ ( e i ) = e i and τ ( a ij ) = a ji for all i, j .Let us describe a corresponding cellular decomposition. First let x := a , x := e , x := a , x := e . and set y i := τ ( x i ), for i ∈ [1 , X (1) := { x } , X (2) := { x , x } , X (3) := { x } , and Y (1) := { y } , Y (2) := { y , y } , Y (3) := { y } , parametrized by the totally ordered sets B := { } , B := { < } , and B := { } ,respectively. We may then define a cellular decomposition Z = J ′ ⊕ J ′ ⊕ J ′ , where J ′ j := span { xy | x ∈ X ( j ) , y ∈ Y ( j ) } , for j ∈ [1 , Λ op denote the set Λ = Λ +3 (1 , ,
1) with the opposite total ordering.Then one may then check using formula (5.20) and the proof of Lemma 6.3 that S Z (1 ,
2) = Γ Z has the cellular basis described in the table below, where λ runsthrough all multipartitions in the set Λ op , and where S , T denote standard multi-tableaux of shape λ , respectively. λ S T z S , T (ø , ø , (2)) (ø , ø , ) (ø , ø , ) e ⊗ (ø , (1) , (1)) (ø , , ) (ø , , ) e ∗ e ′′ (ø , , ) e ∗ a (ø , , ) (ø , , ) e ∗ a ′′ (ø , , ) e ∗ ( a a )(ø , (1 , , ø) (ø , , ø) (ø , , ø) e ∗ ( a a )(ø , (2) , ø) (ø , , ø) (ø , , ø) e ⊗ ′′ (ø , , ø) e ∗ a ′′ (ø , , ø) a ⊗ (ø , , ø) (ø , , ø) e ∗ a ′′ (ø , , ø) e ∗ ( a a ) + a ∗ a ′′ (ø , , ø) a ∗ a a (ø , , ø) (ø , , ø) a ⊗ ′′ (ø , , ø) a ∗ ( a a ) ′′ (ø , , ø) ( a a ) ⊗ ((1) , ø , (1)) ( , ø , ) ( , ø , ) ( a a ) ∗ e ((1) , (1) , ø) ( , , ø) ( , , ø) e ∗ ( a a ) ′′ ( , , ø) a ∗ ( a a )( , , ø) ( , , ø) a ∗ ( a a ) ′′ ( , , ø) ( a a ) ∗ ( a a )((2) , ø , ø) ( , ø , ø) ( , ø , ø) ( a a ) ⊗ The symbol, ø, is used above to denote an empty partition or tableau, respectively,and the symbol ′′ denotes a repeated item from the above entry. ELLULARITY OF GENERALIZED SCHUR ALGEBRAS 27
Cellularity of wreath products A ≀ S d . Let us first recall a result of [16]concerning idempotents fixed by an anti-involution.
Lemma 6.7 ([16]) . Let A be a cellular algebra with anti-involution τ . If e ∈ A is an idempotent fixed by τ , then the algebra eAe is cellular with respect to therestriction of τ . We then have the following consequence of Theorem 6.5, which is obtained viageneralized Schur-Weyl duality.
Corollary 6.8.
Suppose d ∈ N . If A is a cellular algebra, then A ≀ S d is alsocellular.Proof. Fix some n ≥ d . Write S A = S A ( n, d ), and let e ∈ S A denote the idempotent e := ξ ω . It then follows by Proposition 4.2.(ii) that there is an algebra isomorphism A ≀ S d ∼ = eS A e . Since τ ( e ) = ( E , ) tr ∗ · · · ∗ ( E d,d ) tr = e, the cellularity of A ≀ S d follows from Theorem 6.5 and Lemma 6.7. (cid:3) Since the above result holds for an arbitrary cellular algebra A , we thus obtainan alternate proof of the main results of [7] and [11] mentioned in the introduction. References [1] K. Akin, D.A. Buchsbaum and J. Weyman,
Schur functors and Schur complexes , Adv. inMath. (1982), no. 3, 207–278.[2] A.-L. Cauchy, M´emoire sur les fonctions altern´ees et sur les sommes altern´ees, Exercicesd’analyse et de phys. math., ii (1841), 151–159; or Œuvres compl`etes, 2`eme s´erie xii, Gauthier-Villars, Paris, 1916, 173–182.[3] E. Cline, B. Parshall and L. Scott, Finite-dimensional algebras and highest weight categories ,J. Reine Angew. Math. (1988), 85–99.[4] A. Evseev and A. Kleshchev,
Turner doubles and generalized Schur algebras , Adv. Math. (2017), 665–717.[5] ,
Blocks of symmetric groups, semicuspidal KLR algebras and zigzag Schur-Weylduality , Ann. of Math. (2) (2018), no. 2, 453–512.[6] E.M. Friedlander and A. Suslin,
Cohomology of finite group schemes over a field , In-vent. Math. (1997), no. 2, 209–270.[7] T. Geetha and F.M. Goodman,
Cellularity of wreath product algebras and A -Brauer algebras ,J. Algebra (2013), 151–190.[8] J.J. Graham and G.I. Lehrer, Cellular algebras , Invent. Math. (1996), 1–34.[9] J.A. Green, Polynomial Representations of GL n . (Lecture Notes in Math. 830), Springer-Verlag, New York 1980.[10] , Combinatorics and the Schur algebra , J. Pure Appl. Algebra (1993), 89–106.[11] R. Green, Cellular Structure of Wreath Product Algebras , J. Pure Appl. Algebra 224 (2020),no. 2, 819835.[12] M. Hashimoto and K. Kurano,
Resolutions of determinantal ideals: n -minors of ( n +2) -squarematrices , Adv. Math. (1992), no. 1, 1–66.[13] A. Kleshchev and R. Muth, Based quasi-hereditary algebras , arXiv:1810.02844.[14] ,
Generalized Schur algebras , arXiv:1810.02846.[15] ,
Schurifying quasi-hereditary algebras , arXiv:1810.02849.[16] S. K¨onig, and C.C. Xi,
On the structure of cellular algebras , Algebras and modules, II(Geiranger, 1996), 365–386, CMS Conf. Proc., 24, Amer. Math. Soc., Providence, RI, 1998.[17] H. Krause,
Highest weight categories and strict polynomial functors. With an appendix byCosima Aquilino , EMS Ser. Congr. Rep., Representation theory, current trends and perspec-tives, 331–373, Eur. Math. Soc., Z¨urich, 2017.[18] I.G. Macdonald, Symmetric functions and Hall polynomials, second edition, Oxford Math.Mon., (1995).
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