Cellularity of endomorphism algebras of Young permutation modules
aa r X i v : . [ m a t h . R T ] J un Cellularity of endomorphism algebras ofYoung permutation modules.
Stephen Donkin
Department of Mathematics, University of York, York YO10 5DD [email protected]
Abstract
Let E be an n -dimensional vector space. Then the symmetric groupSym( n ) acts on E by permuting the elements of a basis and hence on the r -fold tensor product E ⊗ r . Bowman, Doty and Martin ask, in [1], whetherthe endomorphism algebra End Sym( n ) ( E ⊗ r ) is cellular. The module E ⊗ r isthe permutation module for a certain Young Sym( n )-set. We shall showthat the endomorphism algebra of the permutation module on an arbitraryYoung Sym( n )-set is a cellular algebra. We determine, in terms of thepoint stabilisers which appear, when the endomorphism algebra is quasi-hereditary. We fix a positive integer n . The symmetric group of degree n is denotedSym( n ). For a partition λ = ( λ , λ , . . . ) of n we have the Young subgroup,i.e. the group Sym( λ ) = Sym( λ ) × Sym( λ ) ×· · · , regarded as a subgroup ofSym( n ) in the usual way. By a Young Sym( n )-set we mean a finite Sym( n )-set such that each point stabiliser is conjugate to a Young subgroup. Let R be a commutative ring. Our interest is in the endomorphism algebraEnd Sym( n ) ( R Ω) of the permutation module R Ω on a Young Sym( n )-set Ω.We shall show that End Sym( n ) ( Z Ω) has a cellular structure, Theorem 6.4,hence by base change so has End
Sym( n ) ( R Ω), for an arbitrary commutativering R .Taking the base ring now to be a field k of positive characteristic, we givea criterion for End Sym( n ) ( k Ω) to be a quasi-hereditary algebra, in terms ofthe set of partitions λ of n for which Sym( λ ) occurs as a point stabiliser,and the characteristic p of k , see Theorem 6.4. This is applied to the caseΩ = I ( n, r ), the set of maps from { , . . . , r } to { , . . . , n } , for a positiveinteger r , with Sym( n ) acting by composition of maps. The permutationmodule kI ( n, r ) may be regarded as the r th tensor power E ⊗ r of an n -dimensional vector space E , and we thus determine when End Sym( n ) ( E ⊗ r )is quasi-hereditary, see Proposition 7.3.1ur procedure is to analyse the endomorphism algebra of a Young permu-tation module in the spirit of the Schur algebra S ( n, r ) (which is a specialcase). Of particular importance to us will be the fact that the Schur alge-bras is quasi-hereditary. There are several approaches to this (see e.g. [5,Section A5] and [18]) but for us the most convenient is that of Green, [9].This has the advantage of being a purely combinatorial account carried outover an arbitrary commutative base ring. So we regard what follows as amodest generalisation of some aspects of [9]: we follow Green’s approachand notation to a large extent. We write mod( S ) for the category of finitely generated modules over aring S .Let G be a finite group and K a field of characteristic 0. Let X be afinitely generated KG -module. Suppose that all composition factors of X are absolutely irreducible. Let U , . . . , U d be a complete set of pairwise non-isomorphic composition factors of X . We write X as a direct sum of simplemodules X = X ⊕ · · · ⊕ X h . For 1 ≤ i ≤ d let m i be the number of elements r ∈ { , . . . , h } such that X r is isomorphic to U i . Let S = End G ( X ). Then S is isomorphic to the product of the matrix algebras M m ( K ) , . . . , M m d ( K ). Let the corresponding irreducible modules for S be L , . . . , L d . We have an exact functor from f : mod( KG ) → mod( S ), givenon objects by f ( Z ) = Hom Sym( n ) ( X, Z ). Moreover we have S = f ( X ) = L hr =1 Hom G ( X, X r ). If follows that the modules L i = f U i = Hom G ( X, U i ),1 ≤ i ≤ d , form a complete set of pairwise non-isomorphic irreducible S -modules.The situation in positive characteristic is similar, cf. [8, (3.4) Proposition].Suppose now that F is any field which is a splitting field for G . Let Y be afinitely generated KG -module such that every indecomposable componentis absolutely indecomposable. Let V , . . . , V e be a complete set of pairwisenon-isomorphic indecomposable summands of Y . We write Y as a directsum of indecomposable modules Y = Y ⊕ · · · ⊕ Y k . For 1 ≤ j ≤ e let n j be the number of elements r ∈ { , . . . , k } such that X r is isomorphicto V j . Let T = End G ( Y ). Then each P j = Hom G ( Y, V j ) is naturally a T -module and the modules P , . . . , P e form a complete set of pairwise non-isomorphic projective T -modules. Let N j be the head of P j , 1 ≤ j ≤ e .Then the modules N , . . . , N e form a complete set of pairwise non-isomorphicirreducible T -modules. The dimension of N j over F is n j .We now fix a positive integer n . We write Par( n ) for the set of partitionsof n . By the support ζ (Ω) of a Young Sym( n )-set Ω we mean the set of λ ∈ Par( n ) such that the Young subgroup Sym( λ ) is a point stabiliser. Let2 be a commutative ring. For a Young Sym( n )-set Ω we write S Ω ,R for theendomorphism algebra End Sym( n ) ( R Ω) of the permutation module R Ω. For λ ∈ Par( n ) we write M ( λ ) R for the permutation module R Sym( n ) / Sym( λ ).We have the usual dominance partial order E on Par( n ). Thus, for λ =( λ , λ , . . . ) , µ = ( µ µ , . . . ) ∈ Par( n ), we write λ E µ if λ + · · · + λ a ≤ µ + · · · + µ a for all 1 ≤ a ≤ n .Recall that the Specht modules Sp( λ ) Q , λ ∈ Par( n ), form a complete set ofpairwise irreducible Q Sym( n )-modules. For λ ∈ Par( n ) we have M ( λ ) Q =Sp( λ ) Q ⊕ C , where C is a direct sum of modules of the form Sp( µ ) with λ ⊳ µ , and moreover every Specht module Sp( µ ) Q with λ ⊳ µ occurs in C (see for example [12, 14.1]).For a Young Sym( n )-set Ω we define ζ D (Ω) = { µ ∈ Par( n ) | µ D λ for some λ ∈ ζ (Ω) } . Thus the composition factors of Q Ω are { Sp( µ ) Q | µ ∈ ζ D (Ω) } and, setting ∇ Ω ( λ ) Q = Hom Sym( n ) ( Q Ω , Sp( µ ) Q ), we have the following. Lemma 2.1.
The modules ∇ Ω ( λ ) Q , λ ∈ ζ D (Ω) , form a complete set ofpairwise non-isomorphic irreducible S Ω , Q -modules. Remark 2.2.
Since S Ω , Q is a direct sum of matrix algebras over Q it issemisimple, all irreducible modules are absolutely irreducible and dim Q S Ω , Q = P λ ∈ ζ D (Ω) (dim Q ∇ Ω ( λ ) Q ) . We now let k be a field of characteristic p >
0. For λ ∈ Par( n ) we have theYoung module Y ( λ ) for k Sym( n ), labelled by λ , as described in [5, Section4.4] for example. Then we have M ( λ ) k = Y ( λ ) ⊕ C , where C is a directsum of Young modules Y ( µ ), with λ ⊳ µ , see for example [5, Section 4.4(1) (v)]. A partition λ = ( λ , λ , . . . ) will be called p -restricted (also calledcolumn p -regular) if λ i − λ i +1 < p for all i ≥
1. A partition λ has a uniqueexpression λ = X i ≥ p i λ ( i )where each λ ( i ) is a p -restricted partition. This is called the base p (or p -adic) expansion of λ .We write Λ( n ) for the set of all n -tuples of non-negative integers. Anexpression λ = P i ≥ p i γ ( i ), with all γ ( i ) ∈ Λ( n ) (but not necessarily re-stricted) will be called a weak p expansion.For an n -tuple of non-negative integers γ we write γ for the partitionobtained by arranging the entries in descending order. Definition 2.3.
For λ, µ ∈ Par( n ) we shall say that µ p -dominates λ , andwrite µ D p λ (or λ E p µ ) if there exists a weak p expansion λ = P i ≥ p i γ ( i ) ,such that µ ( i ) D γ ( i ) for all i ≥ , where µ = P i ≥ p i µ ( i ) is the base p expansion of µ . λ E p µ implies λ E µ .By [4, Section 3, Remark], for λ, µ ∈ Par( n ), then module Y ( µ ) appearsas a component of M ( λ ) k if and only if λ E p µ . For a Young Sym( n )-set Ωwe define ζ D p (Ω) = { µ ∈ Par( n ) | µ D p λ for some λ ∈ ζ (Ω) } . Writing P ( µ ) = Hom Sym( n ) ( k Ω , Y ( µ )) and writing L ( µ ) for the head of P ( λ ), for µ ∈ ζ D p (Ω) we have the following. Lemma 2.4.
The modules L ( µ ) , µ ∈ ζ D p (Ω) , form a complete set of pair-wise non-isomorphic irreducible S Ω ,k -modules. We fix a positive integer n and a Young Sym( n )-set Ω. Here we assume thebase ring R is either the ring integers Z or the field of rational numbers Q .We write M Ω ,R , or just M R for the permutation module R Ω over R Sym( n ).We also just write M for M Ω , Z . We shall sometimes write simply S R for S Ω ,R and just S for S Z . We identify S with a subring or S Q in the naturalway.Let { Ø α | α ∈ Λ Ω } be a complete set of orbits in Ω. For λ ∈ ζ (Ω) we pick α ( λ ) ∈ Λ Ω such that Sym( λ ) is a point stabiliser of some element of Ø α .We put M α,R = R Ø α , and sometimes write just M α for M α, Z , for α ∈ Λ Ω . For β ∈ Λ Ω we define the element ξ β of S R to be the projection onto M β,R coming from the decomposition M R = L α ∈ Λ Ω M α,R . Then each ξ α isidempotent and we have the orthogonal decomposition:1 S = X α ∈ Λ Ω ξ α . For a left S R -module V and β ∈ Λ Ω we have the β weight space β V = ξ β V and the weight space decomposition V = M α ∈ Λ Ω α V. For λ ∈ Par( n ) we define λ V = ( ξ α ( λ ) V, if λ ∈ ζ (Ω);0 , otherwise.Similar remarks apply to weight spaces of right S R -modules. Lemma 3.1.
Let λ ∈ ζ D (Ω) . Then(i) dim Q λ ∇ Ω ( λ ) Q = 1 ; and(ii) if µ ∈ Par( n ) and µ ∇ Ω ( λ ) Q = 0 then µ E λ . roof. Let µ ∈ Par( n ) and suppose µ ∇ Ω ( λ ) Q = 0. Thus ξ µ Hom
Sym( n ) ( M Q , Sp( λ ) Q ) = 0 i.e. Hom Sym( n ) ( M ( µ ) Q , Sp( λ ) Q ) = 0 and so µ D λ , giving (ii). Moreover ξ λ Hom
Sym( n ) ( M Q , Sp( λ ) Q ) = Hom Sym( n ) ( M ( λ ) Q , Sp( λ ) Q ) = Q giving (i).For λ ∈ Par( n ) we set ξ λ = ( ξ α ( λ ) , if λ ∈ ζ (Ω) :0 , otherwise.For λ ∈ Par( n ) we set S R ( λ ) = S R ξ λ S R and for σ ⊆ Par( n ) set S R ( σ ) = X λ ∈ σ S R ( λ ) . We also write simply S ( λ ) for S Z ( λ ) and S ( σ ) for S Z ( σ ).Let ≤ be a partial order on Par( n ) which is a refinement of the dominancepartial order. For λ ∈ ζ (Ω) we set S R ( ≥ λ ) = S R ( σ ), where σ = { µ ∈ Par( n ) | µ ≥ λ } , and S R ( > λ ) = S R ( τ ), where τ = { µ ∈ Par( n ) | µ > λ } . Thus S R ( ≥ λ ) = S R ξ λ S R + S ( > λ ) . We set V R ( λ ) = S R ( ≥ λ ) /S R ( > λ ). So we have V R ( λ ) λ = ( S R ξ λ + S R ( > λ )) /S R ( > λ ) , λ V R ( λ ) = ( ξ λ S R + S R ( > λ )) /S R ( > λ )and the multiplication map S R ξ λ × ξ λ S R → S R induces a surjective map φ R ( λ ) : V R ( λ ) λ ⊗ R λ V R ( λ ) → V R ( λ ) . For left S R -modules P, Q and λ ∈ Par( n ) we define Hom λ Sym( n ) ( P, Q ) tobe the R -submodule of Hom Sym( n ) ( P, Q ) spanned by all composite maps f ◦ g , with f ∈ Hom
Sym( n ) ( M ( λ ) R , Q ) and g ∈ Hom
Sym( n ) ( P, M ( λ ) R ). Fora subset σ of Par( n ) we setHom σ Sym( n ) ( P, Q ) = X λ ∈ σ Hom λ Sym( n ) ( P, Q ) . We note some similarity of our approach here via these groups of homo-morphisms with the approach to Schur algebras due to Erdmann, [6] viastratification. 5or λ ∈ Par( n ) we define Hom ≥ λ Sym( n ) ( P, Q ) = Hom σ Sym( n ) ( P, Q ), where σ = { µ ∈ Par( n ) | µ ≥ λ } , and Hom >λ Sym( n ) ( P, Q ) = Hom τ Sym( n ) ( P, Q ), where τ = { µ ∈ Par( n ) | µ > λ } .Note that if λ ζ (Ω) then V R ( λ ) = 0. Suppose λ ∈ ζ (Ω). Then we have S R ξ λ S R = X α,β,γ,δ ∈ Λ Ω Hom
Sym( n ) ( M α,R , M β,R ) ξ λ Hom
Sym( n ) ( M γ,R , M δ,R )= X α,δ ∈ Λ Ω Hom
Sym( n ) ( M α,R , M α ( λ ) ) ξ λ Hom
Sym( n ) ( M α ( λ ) , M δ,R )= M α,β ∈ Λ Ω Hom λ Sym( n ) ( M α,R , M β,R )and hence S R ( σ ) = M α,β ∈ Λ Ω Hom σ Sym( n ) ( M α,R , M β,R ) (1)for σ ⊆ Par( n ). In particular we have S R ( ≥ λ ) = M α,β ∈ Λ Ω Hom ≥ λ Sym( n ) ( M α,R , M β,R )and S R ( > λ ) = M α,β ∈ Λ Ω Hom >λ Sym( n ) ( M α,R , M β,R )and hence V R ( λ ) = M α,β ∈ Λ Ω Hom ≥ λ Sym( n ) ( M α,R , M β,R ) / Hom >λ Sym( n ) ( M α,R , M β,R ) . (2) Example 3.2.
Of crucial importance is the motivating example of the usualSchur algebra S ( n, r ) . Let R be a commutative ring and let E R be a free R -module of rank n . Then Sym( r ) acts on the r -fold tensor product E ⊗ rR = E R ⊗ · · · ⊗ R E R by place permutation, and the Schur algebra S R ( n, r ) maybe realised as End
Sym( r ) ( E ⊗ rR ) .We choose an R -basis e , . . . , e n of E R . We write I ( n, r ) for the set ofmaps from { , . . . , r } to { , . . . , n } . We regard i ∈ I ( n, r ) as an r -tuple ofelements ( i , . . . , i r ) with entries in { , . . . , n } (where i a = i ( a ) , ≤ a ≤ r ).The group Sym( r ) acts on I ( n, r ) composition of maps, i.e. by w · i = i ◦ w − ,for w ∈ Sym( r ) , i ∈ I ( n, r ) . Moreover, for i ∈ I ( n, r ) , w ∈ Sym( r ) , we have w · e i = e i ◦ w − .We may thus regard E ⊗ rR as the R Sym( r ) permutation module R Ω on Ω = I ( n, r ) . Note that ζ (Ω) = Λ + ( n, r ) , the set of partitions of r with atmost n parts. We write Λ( n, r ) for the set of weights, i.e. the set of n -tuplesof non-negative integers α = ( α . . . , α n ) such that α + · · · + α n = r . Anelement i of I ( n, r ) has weight wt( i ) = ( α , . . . , α n ) ∈ Λ( n, r ) , where α a = | i − ( a ) | , for ≤ a ≤ n . For α ∈ Λ( n, r ) we have the orbit Ø α consisting orall i ∈ I ( n, r ) such that wt( i ) = α . Then R Ω = L α ∈ Λ( n,r ) R Ø α . Groups of homomorphisms between Young per-mutation modules
In the situation of the Example 3.2 it follows from the quasi-hereditarystructure of S Z ( n, r ) that V Z ( λ ) is a free abelian group - indeed an explicitbasis is given by Green in [9, (7.3) Theorem, (ii),(iii)]. Thus, taking r = n ,from Section 3, (2), we have the following. Lemma 4.1.
For all λ, µ, τ ∈ Par( n ) the quotient Hom ≥ λ Sym( n ) ( M ( µ ) , M ( τ )) / Hom >λ Sym( n ) ( M ( µ ) , M ( τ )) is torsion free. We can improve on this somewhat. A subset σ of Par( n ) will be calledco-saturated (also said to be a co-ideal) if whenever λ, µ ∈ σ , λ ∈ σ and λ E µ then µ ∈ σ . Proposition 4.2.
Let σ, τ be cosaturated subsets of
Par( n ) with the τ ⊆ σ .Then, for all µ, ν ∈ Par( n ) , the quotient Hom σ Sym( n ) ( M ( µ ) , M ( ν )) / Hom τ Sym( n ) ( M ( µ ) , M ( ν )) is torsion free.Proof. If there is a co-saturated subset θ with τ ⊂ θ ⊂ σ (and θ = σ, τ ) andif Hom σ Sym( n ) ( M ( µ ) , M ( ν )) / Hom θ Sym( n ) ( M ( µ ) , M ( ν ))and Hom θ Sym( n ) ( M ( µ ) , M ( ν )) / Hom τ Sym( n ) ( M ( µ ) , M ( ν ))are torsion free then so isHom σ Sym( n ) ( M ( µ ) , M ( ν )) / Hom τ Sym( n ) ( M ( µ ) , M ( ν )) . Thus we are reduced to the case τ = σ \{ λ } , where λ is a maximal elementof σ . We choose a total order (cid:22) on Par( n ) refining ≤ such that, writingout the elements of Par( n ) in descending order λ ≻ λ · · · ≻ λ h we have τ = { λ , . . . , λ k } , σ = { λ , . . . , λ k +1 } (so λ = λ k +1 ) for some k . Then wehave Hom σ Sym( n ) ( M ( µ ) , M ( ν )) / Hom τ Sym( n ) ( M ( µ ) , M ( ν )= Hom (cid:23) λ Sym( n ) ( M ( µ ) , M ( ν )) / Hom ≻ λ Sym( n ) ( M ( µ ) , M ( ν )which is torsion free by the Lemma.Returning to the general situation we have, by the Proposition and Section3, (2), the following results. Corollary 4.3.
The S -module V ( λ ) is torsion free. Corollary 4.4.
Let σ be cosaturated set (with respect to ≤ ). Then S ( σ ) isa pure submodule of S . Cosaturated
Sym( n ) -sets From Corollary 4.4, if σ is any co-saturated subset of Par( n ) then we mayidentify Q ⊗ Z S ( σ ) with an S Ω , Q -submodule of S Ω , Q via the natural map Q ⊗ Z S ( σ ) → S Q .We now suppose that Ω is cosaturated, by which we mean that ζ (Ω)is a cosaturated subset of Par( n ). We check that much of the structure,described by Green for the Schur algebras in [9], still stands in this moregeneral case.Let σ be a co-saturated subset of the support ζ (Ω) of Ω. Let µ ∈ ζ (Ω). If ∇ Ω ( µ ) Q is a composition factor of S ( σ ) Q then it is a composi-tion factor of S ( λ ) Q and hence of S Q ξ λ , for some λ ∈ σ . Hence we haveHom Sym( n ) ( Sξ λ , ∇ Ω ( µ ) Q ) = 0 and so µ ≥ λ , Lemma 3.1(ii), and therefore µ ∈ σ .We fix λ ∈ ζ (Ω). Then Hom Sym( n ) ( S Q ξ λ , ∇ Ω ( λ ) Q ) = λ ∇ Ω ( λ ) Q = Q , byLemma 3.1(i), so that ∇ Ω ( λ ) Q is a composition factor of S ( ≥ λ ) Q , but notof S ( > λ ) Q . Now we can write S ( ≥ λ ) Q = S ( > λ ) ⊕ I for some ideal I which, as a left S Q -module, has only the composition factor ∇ Ω ( λ ) Q . Hence I is isomorphic to the matrix algebra M d ( Q ), where d = dim ∇ Ω ( λ ) Q , and,as a left S Q -module S ( ≥ λ ) /S ( > λ ) is a direct sum of d copies of ∇ Ω ( λ ) Q .Hence dim Q λ V Q ( λ ) = dim Q Hom
Sym( n ) ( S Q ξ λ , V Q ( λ ))= d dim Q Hom
Sym( n ) ( S Q ξ λ , ∇ Ω ( λ ) Q )= d dim Q λ ∇ Ω ( λ ) Q = d. Thus dim V Q ( λ ) λ ⊗ Q λ V Q ( λ ) = dim V Q ( λ ) and we have:the natural map V Q ( λ ) λ ⊗ Q λ V Q ( λ ) → V Q ( λ ) is an isomorphism. (1)We now consider the integral version. We have the natural surjective map V ( λ ) λ ⊗ Z λ V ( λ ) → V ( λ ). But the rank of V ( λ ) λ is the dimension of V Q ( λ ) λ ,the rank of λ V ( λ ) is the dimension of λ V Q ( λ ), and the rank of V ( λ ) is thedimension of V Q ( λ ) so that, by (1), V ( λ ) λ ⊗ Z λ V ( λ ) and V ( λ ) have the samerank. Thus the surjective map V ( λ ) λ ⊗ Z λ V ( λ ) → V ( λ ) is an isomorphism.We have shown the following. Proposition 5.1.
Assume Ω is cosaturated. Then, for each λ ∈ Par( n ) ,the map V ( λ ) λ ⊗ Z λ V ( λ ) → V ( λ ) induced by multiplication in S , is an isomorphism. Remark 5.2. If k is a field then the corresponding algebras S Ω ,k over k areMorita equivalent to those considered by Mathas and Soriano in [15]. Therethey determined blocks of such algebras (for the Schur algebras themselvesthis was done in [3], and for the quantised case by Cox in [2]). Cellularity of endomorphism algebras of Youngpermutation modules
We now establish our main result, namely that the endomorphism algebraof a Young permutation module has the structure of a cellular algebra. Wefirst recall the notion of a cellular algebra due to Graham and Lehrer, [7].(We have made some minor notational changes to be consistent with thenotation above. The most serious of these is the reversal of the partial orderfrom the definition given in [7].)
Definition 6.1.
Let A be an algebra over a commutative ring R . A celldatum for (Λ + , N, C, ∗ ) for A consists of the following.(C1) A partially ordered set Λ + and for each λ ∈ Λ + a finite set N ( λ ) andan injective map C : ` λ ∈ Λ + N ( λ ) × N ( λ ) → A with image an R -basis of A .(C2) For λ ∈ Λ + and t, u ∈ N ( λ ) we write C ( t, u ) = C λt,u ∈ R . Then ∗ isan R -linear anti-involution of A such that ( C λt,u ) ∗ = C λu,t .(C3) If λ ∈ Λ + and t, u ∈ N ( λ ) then for any element a ∈ A we have aC λt,u ≡ X t ′ ∈ N ( λ ) r a ( t ′ , t ) C λt ′ ,u (mod A ( > λ )) where r a ( t ′ , t ) ∈ R is independent of u and where A ( > λ ) is the R -submoduleof A generated by { C µt ′′ ,u ′′ | µ ∈ Λ + , µ > λ and t ′′ , u ′′ ∈ N ( µ ) } .We say that A is a cellular R -algebra if it admits a cell datum. Let G be a finite group. Let Ω be a finite G -set and let R be a commutativering. Now G acts on Ω × Ω. If
A ⊆ Ω × Ω is G -stable then we have an element a A ∈ End G ( R Ω) satisfying a A ( x ) = X y y where the sum is over all y ∈ Ω such that ( y, x ) ∈ A . We write Orb G (Ω × Ω)for the set of G -orbits in Ω × Ω. Then End RG ( R Ω) free over R on basis a A , A ∈
Orb G (Ω × Ω). We have an involution on Ω × Ω defined by ( x, y ) ∗ = ( y, x ), x, y ∈ Ω. For a G -stable subset A of Ω × Ω we write A ∗ for the G -stable set { ( x, y ) ∗ | ( x, y ∈ Ω } .For A , B ∈
Orb G (Ω × Ω) we have a A a B = X C∈ Orb G (Ω × Ω) n CA , B a C where, for fixed x ∈ A , y ∈ B , the coefficient n CA , B is the cardinality of theset { z ∈ C | ( x, z ) ∈ A and ( z, y ) ∈ B} . It follows that End RG ( R Ω) has aninvolutory anti-automorphism satisfying a ∗D = a D ∗ , for a G -stable subset9 of Ω × Ω. The notion of cellularity has built into it an involutory anti-automorphism ∗ and in the case of endomorphism algebras of permutationmodules, we shall always use the one just defined.We now restrict to the case G = Sym( n ) with Ω a Young Sym( n )-set asusual and label by Ø α , α ∈ Λ Ω , the G -orbits in Ω. Now, for α ∈ Λ Ω and x ∈ Ω we have ξ α ( x ) = ( x, if x ∈ Ø α ;0 , otherwise.Hence ξ α = a A , where A = { ( x, x ) | x ∈ Ø α } and therefore ξ ∗ α = ξ α . Inparticular we have ξ ∗ λ = ξ λ for λ ∈ ζ (Ω). Thus we also have S Ω ,R ( σ ) ∗ = S Ω ,R ( σ ), for σ ⊆ Par( n ).Note that if Γ is a G -stable subset of Ω then we have the idempotent e Γ ∈ S Ω ,R given on elements of Ω by e Γ ( x ) = ( x, if x ∈ Γ ;0 , if x Γ . Thus e Γ = a C where C = { ( y, y ) | y ∈ Γ } and e ∗ Γ = e Γ .So now let Γ be a Young Sym( n )-set and let Ω be a co-saturated YoungSym( n )-set containing Γ. We have the idempotent e = e Γ ∈ S Ω ,R as aboveand S Γ ,R = End Sym( n ) ( R Γ) is naturally identified with eS Ω ,R e . Lemma 6.2.
For λ ∈ ζ (Ω) we have e ∇ Ω ( λ ) Q = 0 if and only if λ ∈ ζ D (Γ) .Proof. We have e = P α ∈ Λ Γ ξ α . Hence e ∇ Ω ( λ ) Q = 0 if and only if ξ α ∇ Ω ( λ ) Q = 0 i.e. P β ∈ Λ Ω ξ α Hom
Sym( n ) ( M β, Q , Sp( λ ) Q ) = 0, for some α ∈ Λ Γ . Hence e ∇ Ω ( λ ) Q = 0 if and only if Hom Sym( n ) ( M β, Q , Sp( λ ) Q ) = 0 forsome β ∈ Λ Γ , i.e. if and only if Hom Sym( n ) ( M ( µ ) Q , Sp( λ )) = 0 for some µ ∈ ζ (Γ), i.e. if and only if there exists µ ∈ ζ (Γ) such that µ E λ .We fix a partial order ≤ on ζ (Ω) refining the partial order E .Let λ ∈ ζ (Ω). We have the section V ( λ ) = S ( ≥ λ ) /S ( > λ ) of S = S Ω .We write J op for the opposite ring of a ring J . We write S env for theenveloping algebra S ⊗ Z S op . We identify an ( S, S )-bimodule with a left S env -module in the usual way.We have the idempotent ˜ e = e ⊗ e ∈ S env and hence the Schur functor˜ f : mod( S env ) → mod(˜ eS env ˜ e ) as in [10, Chapter 6]. Moreover,˜ eS env ˜ e = eSe ⊗ Z ( eSe ) op . Now ˜ f is exact so applying it to the isomorphism V ( λ ) λ ⊗ Z λ V ( λ ) → V ( λ ) of Proposition 5.1 we obtain an isomorphism e V ( λ ) λ ⊗ Z λ V ( λ ) e → eV ( λ ) e (1) . Now ξ λ S + S ( > λ ) = ( Sξ λ + S ( > λ )) ∗ so that eV ( λ ) e = 0 if and only if eV ( λ ) λ = 0. Moreover, V ( λ ) λ is a Z -form of ∇ ( λ ) Q so that eV ( λ ) e = 0 ifand only if e ∇ Ω ( λ ) Q = 0. Hence by, Lemma 6.2,:10 V ( λ ) e = 0 if and only if λ ∈ ζ D (Γ) . (2) . We now assemble our cell data. We have the set Λ + = ζ D (Γ) with partialorder induced from the partial order ≤ on ζ (Ω) (and also denoted ≤ ). Let λ ∈ Λ + . We let n λ = dim Q e ∇ Ω ( λ ) Q and set N ( λ ) = { , . . . , n λ } . Therank of eV ( λ ) λ is n λ . We choose elements d λ, , . . . , d λ,n λ of eSξ λ such thatthe elements d λ, + S ( > λ ) , . . . , d λ,n λ + S ( > λ ) form a Z -basis of eV ( λ ) λ =( eSξ λ + S ( > λ )) /S ( > λ ). Then d ∗ λ, , . . . , d ∗ λ,n λ are elements of ( eSξ λ ) ∗ = ξ λ Se and the elements d ∗ λ, + S ( > λ ) , . . . , d ∗ λ,n λ + S ( > λ ) form a Z -basisof λ V ( λ ) e = ( ξ λ Se + S ( > λ )) /S ( > λ ). Thus d λ,t d ∗ λ,u belongs to eSξ λ Se .We define C : ` λ ∈ Λ + N ( λ ) × N ( λ ) → eSe by C ( t, u ) = C λt,u = d λ,t d ∗ λ,u , for t, u ∈ N ( λ ).Let M be the Z -span of all C λt,u , λ ∈ Λ + , t, u ∈ N ( λ ). We claim that M = eSe . We have S = P λ ∈ Λ Ω Sξ λ S so that if the claim is false then thereexists λ ∈ Λ Ω such that eSξ λ Se M . In that case we choose λ minimalwith this property. First suppose that λ ζ D (Γ). Then we have eV ( λ ) e =0, by (2), i.e., eSξ λ Se ⊆ S ( > λ ) and so eSξ λ Se ⊆ eS ( > λ ) e . However, eS ( > λ ) e = P µ>λ eSξ µ Se ⊆ M , by minimality of λ and so eSξ λ Se ⊆ M .Thus we have λ ∈ Λ + = ζ D (Γ).Now by (1) the map( eSξ λ + S ( > λ )) ⊗ Z ( ξ λ Se + S ( > λ )) → eSξ λ Se + S ( > λ )induced by multiplication is surjective. Moreover we have eSξ λ + S ( > λ ) = P n λ t =1 Z d λ,t + S ( > λ ) and ξ λ Se + S ( > λ ) = P n λ u =1 Z d ∗ λ,u + S ( > λ ) so that eSξ λ Se ⊆ n λ X t,u =1 Z d λ,t d ∗ λ,u + S ( > λ ) = n λ X t,u =1 Z C λt,u + S ( > λ )and hence eSξ λ Se ⊆ n λ X t,u =1 Z C λt,u + eS ( > λ ) e. But now P n λ t,u =1 Z C λt,u ⊆ M by definition and again eS ( > λ ) e ⊆ M by theminimality of λ so that eSξ λ Se ⊆ M and the claim is established.The elements C λt,u , λ ∈ Λ + , 1 ≤ t, u ≤ n λ form a spanning set of eS Ω e = S Γ . But the rank of eSe is the Q -dimension of eS Q e , i.e., the Q -dimension of S Γ , Q and this is P λ ∈ Λ + (dim e ∇ Ω ( λ )) by Remark 2.2. Hence the elements C λt,u , with λ ∈ Λ + , t, u ∈ N ( λ ), form a Z -basis of eSe .We have now checked the defining properties (C1) and (C2) of cell struc-ture and it remains to check (C3). We fix λ ∈ Λ + and let 1 ≤ t, u ≤ n λ . Let a ∈ eSe . Then we have aC λt,u = ad λ,t d ∗ λ,u . P n λ i =1 Z d λ,i + S ( > λ ) = eSξ λ + S ( > λ ) so we may write ad λ,t = P n λ t ′ =1 r a ( t ′ , t ) d λ,t ′ + y for some integers r a ( t ′ , t ) and an element y of S ( > λ ).Thus we have aC λt,u = ad λ,t d ∗ λ,u = n λ X t ′ =1 r a ( t ′ , t ) d λ,t ′ d ∗ λ,u + yd ∗ λ,u = n λ X t ′ =1 r a ( t ′ , t ) C λt ′ ,u + yd ∗ λ,u and hence aC λt,u = n λ X t ′ =1 r a ( t ′ , t ) C λt ′ ,u (mod S ( > λ )) . We have thus checked defining property (C3) and hence proved the fol-lowing.
Theorem 6.3.
Let Γ be a Young Sym( n ) -set. Then (Λ + , N, C, ∗ ) is a cellstructure on S Γ , Z = eS Ω , Z e = End Sym( n ) ( Z Γ) . One now obtains a cell structure on End
Sym( n ) ( R Γ), for any commutativering R by base change.There is also the question of when an endomorphism algebra over a field k is quasi-hereditary. If k has characteristic 0 then End Sym( n ) ( k Γ) is semisim-ple and there is nothing to consider. We assume now that the characteristicof k is p >
0. By [7, Remark 3.10] (see also [13], [14]) End
Sym( n ) ( k Γ) is quasi-hereditary if and only if the number of irreducible End
Sym( n ) ( k Γ)-modules(up to isomorphism) is equal to the length of the cell chain, i.e., | ζ D (Γ) | . ByLemma 2.4 , the number of irreducible End Sym( n ) ( k Γ)-modules is | ζ D p (Γ) | .Moreover, we have ζ D p (Γ) ⊆ ζ D (Γ) and so End Sym( n ) ( k Γ) is quasi-hereditaryif and only if ζ D (Γ) ⊆ ζ D p (Γ). We spell this out in the following result. Theorem 6.4.
Let k be a field of characteristic p > and let Γ be a Young Sym( n ) -set. Then the endomorphism algebra End
Sym( n ) ( k Γ) of the permu-tation module k Γ is quasi-hereditary if and only if for every partition λ of n such that the Young subgroup Sym( λ ) appears as the stabiliser of a pointof Γ and every partition µ D λ there exists a partition τ such that Sym( τ ) appears as a point stabiliser and such that µ p -dominates τ , i.e., there existsa weak p expansion τ = P i ≥ p i γ ( i ) , with γ ( i ) ∈ Λ( n ) , and γ ( i ) E µ ( i ) forall i (where µ = P i ≥ p i µ ( i ) is the base p -expansion of µ and where γ ( i ) isthe partition obtained by writing the parts of γ ( i ) in descending order, for i ≥ ). Remark 6.5.
We emphasise that the above gives a criterion for the endo-morphism algebra
End
Sym( n ) ( k Γ) of the Young permutation module k Γ tobe quasi-hereditary with respect to any labelling of the simple modules by apartially ordered set (which may have nothing to do with those considered bove) thanks to the result of K¨onig and Xi, [14, Theorem 3.]. Thus if Γ doesnot satisfy the condition above then S Γ ,k can not have finite global dimensionby [14, Theorem 3] and hence is not quasi-hereditary. Let R be a commutative ring and let E R be a free R -module on basis e ,R , . . . , e n,R . Let r be a positive integer and let I ( n, r ) be the set describedin Example 3.2. Then the r -fold tensor product E ⊗ rR = E R ⊗ R ⊗ · · · ⊗ R E R has R -basis e i,R = e i ,R ⊗ · · · e i r ,R , i ∈ I ( n, r ), and we thus identify E ⊗ rR with RI ( n, r ), the free R -module on I ( n, r ). Remark 7.1.
The symmetric group
Sym( r ) acts on E ⊗ rR by place permu-tations, i.e. w · e i,R = e i ◦ w − ,R , for w ∈ Sym( r ) , i ∈ I ( n, r ) . Thus wemay regard E ⊗ rR as the permutation module RI ( n, r ) , with Sym( r ) , actingon I ( n, r ) by w · i = i ◦ w − . The endomorphism algebra End
Sym( r ) ( E ⊗ rR ) isthe Schur algebra S R ( n, r ) .The stabiliser of i ∈ I ( n, r ) is the direct product of the symmetric groupson the fibres of i (regarded as a subgroup of Sym( r ) in the usual way). Hence I ( n, r ) is a Young Sym( r ) -set. Hence E ⊗ rR is a Young permutation moduleand hence S R ( n, r ) is cellular. Moreover, ζ ( I ( n, r )) is the set Λ + ( n, r ) of allpartitions of r with at most n parts. This is a co-saturated set and hencefor a prime p we have ζ ( I ( n, r )) = ζ D ( I ( n, r )) = ζ D p ( I ( n, r )) . Hence, for afield k of characteristic p the Schur algebra S k ( n, r ) is quasi-hereditary.However, this is not a new proof since our treatment relies crucially on adetail from Green’s analysis of S Z ( n, r ) as in [9], at least in the case n = r .(See Example 3.2 above and the proofs of the results of Section 4.) We now regard E R as an R Sym( n )-module with Sym( n ) permuting thebasis e ,R , . . . , e n,R in the natural way. This action induces an action on thetensor product E ⊗ rR . Specifically, we have w · e i,R = e w ◦ i,R , for w ∈ Sym( n ), i ∈ I ( n, r ), and we thus regard E ⊗ rR as the permutation module RI ( n, r ). For w ∈ Sym( n ), i ∈ I ( n, r ) we have w ◦ i = i if and only if w acts as the identityon the image of i , so that the stabiliser of i is the group of symmetries ofthe complement of the image of i in { , . . . , n } , identified with a subgroupof Sym( n ) in the usual way. Thus I ( n, r ) is a Young Sym( n )-set so we havethe following consequence of Theorem 6.3, answering a question raised in[1]. Proposition 7.2.
The endomorphism algebra
End
Sym( n ) ( E ⊗ rR ) = End Sym( n ) ( RI ( n, r )) is a cellular algebra. The support of I ( n, r ) consists of hook partitions, more precisely we have ζ ( I ( n, r )) = { ( a, b ) | a + b = n, ≤ b ≤ r } . ζ D ( I ( n, r )) = { λ = ( λ , λ , . . . ) ∈ Par( n ) | λ ≥ n − r } . Let k be a field of characteristic p >
0. Then End
Sym( n ) ( E ⊗ rk ) is quasi-hereditary if and only if ζ D ( I ( n, r )) ⊆ ζ D p ( I ( n, r ), i.e., if and only for every µ = ( µ , µ , . . . ) ∈ Par( n ) with µ ≥ n − r there exists some λ = ( a, b ),1 ≤ b ≤ r , such that λ E p µ .We are able to give an explicit list of quasi-hereditary algebras arising inthe above manner. Proposition 7.3.
Let k be a field of characteristic p > . Let n be apositive integer and E an n -dimensional k -vector space with basis e , . . . , e n . We regard E as a k Sym( n ) -module with Sym( n ) permuting the basis in theobvious way. For r ≥ we regard the r th tensor power E ⊗ r as a k Sym( n ) -module via the usual tensor product action. Then End
Sym( n ) ( E ⊗ r ) is quasi-hereditary if and only if:(i) p does not divide n ; and(ii) either n < p (and r is arbitrary) or n > p and r < p .Proof. We see this in a number of steps. We regard E ⊗ r as the permu-tation module kI ( n, r ), as above, with Sym( n ) action by w · i = w ◦ i , for w ∈ Sym( n ), i ∈ I ( n, r ). We shall say that I ( n, r ) is quasi-hereditary ifEnd Sym( n ) ( E ⊗ r ) is. Step 1. If p divides n then I ( n, r ) is not quasi-hereditary.We have ( n − , ∈ ζ ( I ( n, r )) and ( n, D ( n − ,
1) so that ( n, ∈ ζ D ( I ( n, r )). Now n = pm , for some positive integer m , so that µ = ( n,
0) = p ( m,
0) has base p expansion ( n,
0) = P i ≥ p i µ ( i ), with restricted part µ (0) = 0. Thus if τ = ( a, b ) has weak p -expansion τ = P i ≥ p i γ ( i )and γ ( i ) E µ ( i ), for all i , then γ (0) = 0 and τ is divisible by p . How-ever, this is not the case so no such weak p -expansion exists and µ ∈ ζ D ( I ( n, r )) \ ζ D p ( I ( n, r )). Thus ζ D ( I ( n, r )) = ζ D p ( I ( n, r )) and I ( n, r ) is notquasi-hereditary. Step 2. If p does not divide n then I ( n,
1) is quasi-hereditary.We have ζ ( I ( n, { ( n − , } . If µ ∈ ζ D ( I ( n, \ ζ D p ( I ( n, r )) then µ = ( n, n has base p expansion n = P i ≥ p i n i , with 0 ≤ n i < p for all i ≥ n = 0 and µ has base p expansion µ = P i ≥ p i µ ( i ), with µ ( i ) = ( n i , i ≥ τ = ( n − ,
1) = ( n − ,
1) + X i ≥ p i ( n i , τ has weak p -expansion τ = P i ≥ p i γ ( i ), with γ (0) = ( n − , γ ( i ) =( n i ,
0) for i ≥
1. Moreover γ ( i ) ≤ µ ( i ), for all i so that ( n, ∈ ζ D p ( I ( n, ζ D ( I ( n, ζ D p ( I ( n, I ( n,
1) is quasi-hereditary.
Step 3. If µ ∈ ζ D ( I ( n, r )) is p -restricted then µ ∈ ζ D p ( I ( n, r ))We have µ D ( a, b ) for some n = a + b , 1 ≤ b ≤ r . The partition µ hasbase p expansion µ = P i ≥ p i µ ( i ), with µ ( i ) = 0 for all i ≥ τ = ( a, b ) has week p -expansion τ = P i ≥ p i γ ( i ), with γ (0) =( a, b ) and γ ( i ) = 0 for all i ≥
1. Furthermore we have γ ( i ) E µ ( i ) for all i ≥ µ ∈ ζ D p ( I ( n, r )). Step 4. If n < p then I ( n, r ) is quasi-hereditary.This follows from Step 3 all since elements of Par( n ) are restricted. Step 5. If p < n < p then I ( n, r ) is quasi-hereditary.For a contradiction suppose not and let µ = ( µ , µ , . . . ) ∈ ζ D ( I ( n, r )) \ ζ D p ( I ( n, r )). We have µ D ( a, b ) for some a, b with n = a + b , 1 ≤ b ≤ r . Choose a, b with this property with b ≥ b = 1 then µ ∈ ζ D ( I ( n, ζ D p ( I ( n, b ≥ µ = a . Since µ D ( a, b ) the length l , say, of µ is at mostthe length of ( a, b ), i.e. b + 1. Put ξ = ( ξ , ξ , . . . ) = ( a + 1 , b − ). If µ > a then µ ≥ ξ and, for 1 < i ≤ l , we have µ + · · · + µ i ≥ a + 1 + ( i −
1) = a + i = ξ + · · · + ξ i . So µ D ξ = ( a + 1 , b − ), which is a contradiction, and the claim is estab-lished.Note that µ is non-restricted, by Step 3, and, since µ is a partition of n < p in the base p expansion µ = P i ≥ p i µ ( i ) of µ , we must have µ (1) =(1 ,
0) and µ ( i ) = 0 for i ≥
2. Let τ = ( a, b ). Then τ E µ implies that τ − ( p, E µ − ( p,
0) = µ (0). But now τ = ( a, b ) = ( a − p, b ) + p (1 , p expansion τ = P i ≥ p i γ ( i ) with γ (0) = ( a − p, b ), γ (1) = (1 ,
0) and γ ( i ) = 0 for i >
1. Since γ ( i ) E µ ( i ) for all i ≥ a, b ) E p µ and so µ ∈ ζ D p ( I ( n, r )), a contradiction. Step 6. If n > p and r ≥ p then I ( n, r ) is not quasi-hereditary.Note that ζ ( I ( n, r )) contains ( n − p, p ) and hence ζ D ( I ( n, r )) contains µ = ( n − p, p ). Now we have µ = ( n − p,
0) + p (1 ,
1) and so µ = µ (0) + pξ ,where µ (0) has at most one part and ξ has two parts. Hence in the base p expansion µ = P i ≥ p i µ ( i ), there is for some j ≥
1, such that µ ( j ) has twoparts. 15ow if µ ∈ ζ D p ( I ( n, r )) there there exists some τ = ( a, b ) with weak p expansion τ = P i ≥ p i γ ( i ) such that γ ( i ) E µ ( i ) for all i ≥
0. But then γ ( j ) must have at least two parts. Since j ≥
1, the partition τ = ( a, b ) hastwo parts of size at least p . This is not the case so there is no such weak p expansion and µ ζ D p ( I ( n, r )). Thus ζ D ( I ( n, r )) = ζ D p ( I ( n, r )) and I ( n, r )is not quasi-hereditary. Step 7. If n > p , if p does not divide n and if r < p , then I ( n, r ) isquasi-hereditary.If not there exists µ = ( µ , µ , . . . ) ∈ ζ D ( I ( n, r )) \ ζ D p ( I ( n, r )). Thus µ D ( a, b ), for some n = a + b , b ≥ a, b )with b minimal. Again, by Step 2, we have b ≥ µ = a . If not, we get µ D ( a + 1 , b − ) as in Step 5,contradicting the minimality of b .Thus we have µ + · · · + µ n = n − µ = b < p , in particular we have µ i < p for all i ≥
1. Hence in the base p expansion µ = P i ≥ p i µ ( i ), for all i ≥ µ ( i ) = ( c i , , . . . , ≤ c i < p . Also, µ (0) = ( k, µ , . . . , µ n ),for some k > τ = ( a, b ) = ( k + X i ≥ p i c i , b ) = ( k, b ) + X i ≥ p i ( c i , , . . . , . Thus we have the weak p -expansion τ = P i ≥ p i γ ( i ), with γ (0) = ( k, b )and γ ( i ) = ( c i , , . . . , i ≥
1. Furthermore, γ ( i ) E µ ( i ), for all i ≥ µ ∈ ζ D p ( I ( n, r )) and therefore ζ D ( I ( n, r )) = ζ D p ( I ( n, r )) and I ( n, r ) isquasi-hereditary.Let k be a field. Recall that, for δ ∈ k , and r a positive integer we havethe partition algebra P r ( δ ) over k . One may find a detailed account of theconstruction and properties of P r ( δ ) in for example the papers by Paul P.Martin, [16], [17], and [11], [1]. Suppose now that k has characteristic p > δ = n k , for some positive integer n . Let E n be an n -dimensional vec-tor space with basis e , . . . , e n . Then P r ( n ) = P r ( n k ) acts on E ⊗ rn . Bya result of Halverson-Ram, [11, Theorem 3.6] the image of the represen-tation P r ( n ) → End k ( E ⊗ rn ) is End Sym( n ) ( E ⊗ r ). Moreover, for n ≫ P r ( n ) is faithful. Let N = n + ps , for s suitably large, so that P r ( n ) = P r ( N ) acts faithfully on E ⊗ rN . Thus P r ( n ) is quasi-hereditary if andonly if End Sym( N ) ( E ⊗ rN ) is faithful. Hence from Proposition 7.3 we have thefollowing, which is a special case of a result of K¨onig and Xi, [14, Theorem1.4]. Corollary 7.4.
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