Double Centralisers and Annihilator Ideals of Young Permutation Modules and Signed Young Permutation Mod- Ules

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ΦB : B ⊗A AnnR(P ) → AnnRB (PB).

For an A-algebra B we write Bop for the opposite algebra. We also consider base change on endomorphism algebras. We have the natural map of A-algebras op ΨB : B ⊗A EndR(P ) → EndRB (PB) ′ ′ ′ satisfying ΨB(b ⊗ α)(b ⊗ p)= b b ⊗ α(p), for b, b ∈ B, α ∈ EndR(P ), p ∈ P . We are interested in cases in which ΦB and ΨB are isomorphisms. Our main focus is the case in which R is a group algebra of a finite group and P is a permutation module (and the Hecke algebra analogue of this situation). We take A = Z, G a finite group, X a finite G-set, R = ZG and P = ZX. In this case the endomorphism algebra Λ = EndZG(ZX) has a basis given by orbits of G on X × X and this gives rise to a basis over any coefficient ring. We may then study base change for P regarded as a module over Λ, and hence study the double centraliser of the permutation module over a group algebra. Our primary concern is to show that certain permutation modules for group algebras of symmetric groups (and their quantised analogues over Hecke algebras) have the double centraliser property. There are two aspects to this. We prove this over fields by using connections with the polynomial representation theory of general linear groups (and their quantised analogues). The result is deduced over the integers and then over arbitrary rings by elementary general considerations. We deal with these general elementary considerations in the first part of the paper. They are closely connected with question of whether the annihilator of a module P as above is stable under base change, and the arguments used in both cases are extremely similar. We assume, where convenient that A is a Dedekind domain. We convert the problems into matrix problems for A-modules and show that these are compatible with localisation. In this way we are reduced to the case in which A is a discrete valuation ring. The main result in the first part (see Corollary 2.13 and Proposition 3.3) is that, under certain assumptions, if base change holds from the regular module A to the module A/m and the double centraliser property holds for the R/mR-module P/mP , for each maximal ideal m,

2 then P has the double centraliser property and the base change property on ideals and endomorphism algebras. In particular the double centraliser property holds over all A-algebras obtained by base change from R (so over all coefficient rings if A = Z) if it holds over the residue fields. The second part of the paper is more technical. We establish the double centraliser property for many tilting modules for the quantised general linear groups (equivalently for the q Schur algebras). This information is then transferred to the setting of representations of type A Hecke algebras via the Schur functor and we deduce the double centraliser property and base change on double centralisers and annihilators for many Young permutation modules, including those relevant to the partition algebras. In the penultimate section we give an example of a Young permutation module that does not have the double centraliser property or base change on annihilator ideals. In the final section we make some remarks on cell ideals and annihilators in the spirit of [12], [9], [2] and show that for many Young permutation modules the annihilator ideal is a cell ideal.

Part I: Base Change

2 Localisation and base change for annihilator ideals and centraliser algebras.

All modules will be left unless otherwise specified. Let A be a Dedekind domain or a field, let Λ be an A-algebra and P a Λ-module. We assume that Λ and P are finitely generated and free over A. We fixan A-basis λ1, . . . , λn of Λ and an A-basis p1,...,pd of P . (In the applications, P will be a module over an A-algebra R and P will be regarded as a module for Λ = EndR(P ).) t We have the structure constants cis ∈ A defined by the equations

d t λips = cispt Xt=1 for 1 ≤ i ≤ n, 1 ≤ s ≤ d. Let B be an A-algebra. An element q of ΛB = B ⊗A Λ may be uniquely n written in the form q = i=1 bi ⊗ λi. The condition for q to belong to the annihilator AnnΛB (PB ) of PB = B ⊗A P is that q(b ⊗ ps) = 0, for all b ∈ B, d n P t 1 ≤ s ≤ d, i.e. t=1 i=1 bib ⊗ cispt = 0, i.e. P P n t cisbi = 0 (1) Xi=1 3 for all 1 ≤ s,t ≤ d. It is convenient to generalise this condition to an arbitrary A-module. For a module M and r ≥ 0 we write M r or M (r) for M ⊕···⊕ M, the direct sum of r copies of M.

Deﬁnition 2.1. Let M be an A-module. We shall say that (m1,...,mn) ∈ M n is an admissible vector if

n t cismi = 0 (2) Xi=1 for all 1 ≤ s,t ≤ d.

n n We write Mad for the A-submodule of M consisting of all admissible vectors. n n We have the natural A-module isomorphism A ⊗A M → M , which n n restricts to an A-module homomorphism φM : Aad ⊗A M → Mad. Moreover, n n n n Aad is a pure submodule of A , i.e. A /Aad is torsion free, and hence ﬂat, so n that φM is injective. We shall call an element of M a realisable vector (or n vector-realisable) if it belongs to the image of φM , i.e. y ∈ M is realisable n if for some r ≥ 1 there exist elements (ai1, . . . , ain) ∈ Aad , for 1 ≤ i ≤ r, r n and m1,...,mr ∈ M such that y = i=1(ai1mi, . . . , ainmi). We write Mreal for the submodule of M n consisting of realisable vectors and say that M is n n P vector-realisable if Mad = Mreal (i.e. if φM is surjective). n Remark 2.2. One may give a basis free interpretation of Mad as follows. We have the representation ρ : Λ → EndA(P ) and hence the induced map θ : Λ⊗A M → EndA(P )⊗A M. The A-module EndA(P ) has basis est, where t est(pu) is ps if u = t and 0 otherwise. Then ρ(λi) = s,t cisets and hence, for h = λ ⊗ m ∈ Λ ⊗ M, we have θ(h)= ct e ⊗ m and hence i i i A i,s,t Pis ts i θ(h) = 0 if and only if n ct m = 0, for all s,t. In this way M n is P i=1 is i P ad identiﬁed with the kernel of the natural map θ : Λ ⊗ M → End (P ) ⊗ M. P A A A We study the the situation for endomorphism algebras in a similar fashion.

An element σ ∈ EndΛB (PB) may be represented by a d × d - matrix (σst), with entries in B, deﬁned by the equations

d σ(1 ⊗ ps)= σts ⊗ pt Xt=1 for 1 ≤ s ≤ d. Given such a matrix (σst) we may deﬁne a B-endomorphism as above. The condition for σ to be a ΛB-endomorphism is that it commutes with the action of Λ, i.e. that σ((1 ⊗ λi)(1 ⊗ ps)) = (1 ⊗ λi)σ(1 ⊗ ps) i.e. t u t,u cisσut ⊗ pu = t,u citσts ⊗ pu, i.e.

P P t u cisσut = citσts (3) t t X X 4 for all i, s, u. It will be convenient to generalise this condition to A-modules. For an A-module M we write Matd(M) for the additive group of d × d matrices with entries in M. We regard Matd(M) as an A-module with entrywise action, i.e. a(mst) = (amst). Note that we have the A-module isomorphism Matd(A)⊗A M → Matd(M), taking (ast)⊗m to (astm), for (ast) ∈ Matd(A), m ∈ M.

Deﬁnition 2.3. Let M be an A-module. We shall say that (mst) ∈ Matd(M) is an admissible matrix if

t u cismut = citmts (4) t t X X for all i, s, u.

We write Matd(M)ad for the set of admissible matrices over M. Note that Matd(M)ad is an A-submodule of Matd(M). Note also that the isomorphism Matd(A) ⊗A M → Matd(M) induces a map ψM : Matd(A)ad ⊗ M → Matd(M)ad. Moreover, Matd(A)ad is a pure submodule of Matd(A), i.e. Matd(A)/Matd(A)ad is torsion free, and hence ﬂat, so that ψM is injective. We write Matd(M)real for the image of ψM . We say that an element (mst) of Matd(M) is a realisable matrix (or matrix- realisable) if it belongs to the image of ψM , i.e, if for some r ≥ 1 there i exist elements (ast) ∈ Matd(A)ad, and mi ∈ M, for 1 ≤ i ≤ r, such that r i mst = i−=1 astmi, for all 1 ≤ s,t ≤ d. We shall say that M is matrix- realisable if Mat (M) = Mat(M) (i.e. if ψ is surjective). P d ad real M

Remark 2.4. One may give a basis free interpretation of Matd(M)ad as follows. The endomorphism algebra EndA(P ) is naturally a Λ-bimodule. Hence EndA(P ) ⊗A M is naturally a Λ-bimodule with action λ(φ ⊗ m)µ = λφµ ⊗ m, for λ, µ ∈ Λ, m ∈ M, φ ∈ EndA(P ). The A-module EndA(P ) has basis est, where est(pu) is ps if u = t and 0 otherwise. For 1 ≤ i ≤ n, 1 ≤ u t s,t ≤ d, we have λiest = u ciseut and estλi = v civesv and it follows that an element e ⊗m of End (P )⊗ M belongs to the zeroth Hochschild s,t st st P A A P cohomology group H0(Λ, End (P ) ⊗ M) if and only if P A A t u cismut = citmts t t X X for all i, s, u. In this way we may identify Matd(M)ad and 0 H (Λ, EndA(P ) ⊗A M). 0 We have EndΛ(P )= H (Λ, EndA(P )), hence the induced map 0 EndΛ(P ) ⊗A M → H (Λ, EndA(P ) ⊗A M), and the question is whether this is an isomorphism.

5 n n Remark 2.5. We have the natural isomorphisms Aad ⊗A A → Aad and Matd(A)ad ⊗A A → Matd(A)ad so that the left regular module A is vector- realisable and matrix-realisable. Moreover, a direct sum M = i∈I Mi is vector-realisable (resp. matrix-realisable) if and only if each M is vector- iL realisable (resp. matrix-realisable). In particular a free A-module is vector- realisable and matrix-realisable and if A is a ﬁeld then every A-module is vector-realisable and matrix-realisable.

Lemma 2.6. Let 0 → L → M → N → 0 be a short exact sequence of A-modules. If L and N are vector-realisable (resp. matrix-realisable) then so is M.

Proof. We give the matrix version and leave the vector version to the reader. We may assume that L is a submodule of M and N = M/L. Let (mst) ∈ Matd(M)ad. We set mst = mst + L ∈ M/L, 1 ≤ s,t ≤ d. Then (mst) ∈ i Matd(N)ad. Hence there exist a positive integer u, (ast) ∈ Matd(A)ad and u i ni ∈ N, for 1 ≤ i ≤ u, such that mst = i=1 astni, for all s,t. We write u i ni = mi + L, for some mi ∈ M, 1 ≤ i ≤ u. Then we have mst ≡ i=1 astmi u i P (mod L). Putting lst = mst − i=1 astmi, we have that (lst) ∈ Matd(N)ad j P and hence there exists a positive integer v, (bst) ∈ Matd(A)ad and lj ∈ L, P v j for 1 ≤ j ≤ v, such that lst = j=1 bstlj, for all s,t. Then we have mst = u i v j i=1 astmi + j=1 bstlj, whichP shows that (mst) is realisable. PThe constructionsP introduced above behave well with respect to localisation. Let S be a multiplicatively closed subset of A (containing 1). Then −1 −1 −1 λi we have the S A-algebra S Λ, with S A basis 1 , 1 ≤ i ≤ n, and the −1 −1 ps S P -module with S A-basis 1 , 1 ≤ s ≤ d. We consider admissible vectors and matrices for an S−1A-module with respect to these bases and the t cis structure constants 1 . Lemma 2.7. Let S be a multiplicatively closed subset of A and let M be an A-module. n n n (i) The inclusions of Mad and Mreal into M induce isomorphisms

−1 n −1 n S Mad → (S M)ad and −1 n −1 n S Mreal → (S M)real.

(ii) The inclusions of Matd(M)ad and Matd(M)real into Matd(M) induce isomorphisms

−1 −1 S Matd(M)ad → Matd(S M)ad and −1 −1 S Matd(M)real → Matd(S M)real. (iii) If M is a vector-realisable (resp. matrix-realisable) A-module then S−1M is a vector-realisable (resp. matrix-realisable) S−1A-module.

6 For an A-module M and a maximal ideal m of A we write Mm for the localisation S−1M, where S = A\m.

Proposition 2.8. Let M be an A-module. Then M is a vector-realisable (resp. matrix-realisable) A-module if and only if Mm is a vector-realisable (resp. matrix-realisable) Am-module for every maximal ideal m of A.

Proof. We give the matrix version. If M is matrix-realisable A-module and m is a maximal ideal of A then Mm is a matrix-realisable Am-module by Lemma 2.7(iii). Now suppose that for every maximal ideal m of A the Am-module Mm is matrix-realisable. Let N = Matd(M)ad/Matd(M)real. The short exact sequence 0 → Matd(M)real → Matd(M)ad → N → 0 gives rise to an exact sequence

0 → (Matd(M)real)m → (Matd(M)ad)m → Nm → 0 and hence, by Lemma 2.7(ii), an exact sequence

0 → Matd(Mm)real → Matd(Mm)ad → Nm → 0.

Thus we have Nm = 0 for every maximal ideal m of A and hence, by [1, Proposition 3.8], N = 0, i.e. Matd(M)ad = Matd(M)real. We obtain the following criterion for all A-modules to be realisable in terms of residue ﬁelds.

Proposition 2.9. If A/m is a vector-realisable (resp. matrix-realisable) A- module, for every maximal ideal m of A, then every A-module is vector- realisable (resp. matrix-realisable).

Proof. As usual we give the matrix version. By Proposition 2.8 and Remark 2.5, we may assume that A is local, i.e. A is a discrete valuation ring, with maximal ideal m, say. Let M be an A-module and let (mst) ∈ Matd(M)ad. We let M0 be the A-submodule generated by the elements mst, 1 ≤ s,t ≤ d. If M0 is matrix-realisable then the matrix (mst) is realisable. Thus we may assume that M is finitely generated. Let T (M) be the torsion submodule of M. Then M/T (M) is a free A- module and so is matrix-realisable by Remark 2.5. Hence, by Lemma 2.6, it suffices to prove that T (M) is matrix-realisable, i.e. we may assume that M is a finitely generated torsion module. We choose a positive integer k such that mkM = 0. If mi−1M/miM is matrix-realisable, for 1 ≤ i ≤ k, then so is M, by Lemma 2.6. So we may assume mM = 0, and so M is an A/m vector space. Hence M is matrix- realisable as an F = A/m-module.

7 Consider (mst) ∈ Matd(M)ad. Then, for some u there exists admissible i matrices (fst), with entries in F , and mi ∈ M, for 1 ≤ i ≤ u, such that u i mst = i=1 fstmi, for all 1 ≤ s,t ≤ d. By hypothesis F is a matrix- realisable A-module, so that, for 1 ≤ i ≤ r, there exists an integer vi and P ij admissible matrices (ast) (with entries in A) and gij ∈ F , for 1 ≤ j ≤ vi, i vi ij such that fst = j=1 astgij, for 1 ≤ i ≤ r. Hence, for all 1 ≤ s,t ≤ d, we have P u i ij ij mst = fstmi = astgijmi = astxij Xi=1 Xi,j Xi,j where xij = gijmi, which shows that (mst) is realisable, and completes the proof.

Remark 2.10. Suppose that φ : A → F is a homomorphism from A to a field F and K is a field extension. We note that F is vector-realisable (resp. matrix-realisable) if and only if K is. We shall deal with the matrix version. We choose a basis by, y ∈ Y , of K as an F -space, such that by0 = 1 for some y0 ∈ Y . Suppose that F is matrix realisable and let (kst) ∈ Matd(K)ad. Then for some finite subset Y1 of Y we have, for all 1 ≤ s,t ≤ d, equations

y kst = fstby yX∈Y1 y y for some fst ∈ F . Moreover, for each y ∈ Y1, the matrix (fst) belongs to Matd(F )ad. Hence, for each y ∈ Y1, there exists a positive integer hy, yi yi elements αst ∈ F , such that, for ﬁxed y and i the matrix (αst) belongs to Matd(F )ad, and elements xyi ∈ K, for 1 ≤ i ≤ hy, such that

hy y yi fst = αstxyi. Xi=1 Hence we have

yi yi kst = αstxyiby = αsttyi y∈Y1X,1≤i≤hy y∈Y1X,1≤i≤hy where tyi = xyiby, which shows that (kst) is realisable, and hence K is a matrix-realisable A-module. Conversely suppose that K is a realisable A-module. We consider (fst) ∈ Matd(F )ad. Then (fst) ∈ Matd(K)ad. Hence there exists a positive integer i h, elements (ast) of Matd(A)ad, and k1,...,kh ∈ K such that

h i fst = astki Xi=1

8 for 1 ≤ s,t ≤ d. We write ki = y∈Y βiyby, as an F -linear combination of the basis elements b , for 1 ≤ i ≤ h. Thus we get y P i fst = astβiyby i=1,...,h,yX ∈Y and so, equating coeﬃcients of yb0 = 1 we get

h i fst = astβiy0 Xi=1 for 1 ≤ s,t ≤ d, which shows that (fst) is realisable and hence F is a realisable A-module. It follows that, in the context of Proposition 2.9, every A-module is vector- realisable (resp. matrix-realisable) provided that, for each maximal ideal m of A there is a ﬁeld F containing A/m which is vector-realisable (resp. matrix realisable) as an A-module.

Remark 2.11. Suppose that A = F [t,t−1] is the ring of Laurent polynomials over a ﬁeld F . Let K be a ﬁeld extension of F . Then we obtain, by base −1 change AK = K[t,t ], the AK-algebra ΛK = K ⊗F Λ and the ΛK-module PK = K ⊗F P . Then by arguments similar to those of the above remark, we see that an A-module M is vector-realisable (resp. matrix realisable) if and only if the AK-module MK = K ⊗F M is a vector-realisable (resp. matrix realisable) AK -module. We leave the details to the interested reader. Remark 2.12. Let B be an A-algebra. The natural map

ΦB : B ⊗A AnnΛ(P ) → AnnΛB (PB) admits a factorisation of A-module maps

α n β n γ B ⊗A AnnΛ(P ) −→ Aad ⊗A B −→ Bad −→ AnnΛB (PB)

n where α : B ⊗A AnnΛ(P ) → Aad ⊗A B satisfies n α(b ⊗ i=1 aiλi) = (a1, . . . , an) ⊗ b, for b ∈ B, i aiλi ∈ AnnΛ(P ), where n n β : Aad ⊗A B → Bad satisfies β((a1, . . . , an) ⊗ b) = (a1b, . . . , anb), for P n Pn (a1, . . . , an) ∈ Aad, b ∈ B, and where γ : Bad → AnnΛB (PB ) satisfies n n γ(b1, . . . , bn)= i=1 bi ⊗ λi, for (b1, . . . , bn) ∈ Bad. Moreover, the maps α and γ are isomorphisms so that Φ is an iso- P B morphism if and only if β is an isomorphism i.e. if and only if B is a vector-realisable A-module. Likewise, the natural map

op ΨB : B ⊗A EndΛ(P ) → EndΛB (PB )

9 admits a factorisation of A-module maps

op δ ǫ η B ⊗A EndΛ(P ) −→ Matd(A)ad ⊗A B −→ Matd(B)ad −→ EndΛB (PB)

op where δ : B ⊗A EndA(P ) → Matd(A)ad ⊗A B satisﬁes δ(b ⊗ σ) = (σst) ⊗ b, for b ∈ B, σ = (σst) ∈ EndA(P ), where ǫ : Matd(A)ad ⊗A B → Matd(B)ad satisﬁes ǫ((σst) ⊗ b) = (σstb), for (σst) ∈ Matd(A)ad, b ∈ B, and where

η : Matd(B)ad → EndΛB (PB ) satisﬁes η((bst))(b ⊗ pu) = t bbtu ⊗ pt, for (b ) ∈ Mat (B) , b ∈ B, 1 ≤ u ≤ d. st d ad P Moreover, the maps δ and η are isomorphisms so that ΨB is an isomorphism if and only if ǫ is, i.e. if and only if B is a matrix realisable A-module.

From Proposition 2.9 and Remark 2.10 we thus get the following result.

Corollary 2.13. Suppose that Λ is an A-algebra which is free of finite rank over A and that P is a Λ-module which is free of finite rank over A. (i) Suppose that for every maximal idea m there exists a homomorphism from A to a field F with kernel m such that natural map F ⊗A AnnΛ(P ) →

AnnΛF (PF ) is surjective. Then for any A-algebra B the natural map

ΦB : B ⊗A AnnΛ(P ) → AnnΛB (PB) is an isomorphism. (ii) Suppose that for every maximal idea m there exists a homomorphism from A to a ﬁeld F with kernel m such that natural map F ⊗A EndΛ(P ) →

EndΛF (PF ) is surjective. Then for any A-algebra B the natural map

op ΨB : B ⊗A EndΛ(P ) → EndΛB (PB ) is an isomorphism.

3 Double Centralisers

We continue to assume that A is a Dedekind domain or a ﬁeld, that R is an A-algebra and that P an R module.

Hypotheses: We shall require the following properties: (i) the A-algebra R and the R-module P are free of ﬁnite rank over A; (ii) EndR(P ) is free of ﬁnite rank over A; and (iii) EndR(P ) has the base change property on endomorphism algebras, i.e. op the map ΨB : B ⊗A EndR(P ) → EndRB (PB ) is an isomorphism, for every A-algebra B.

When these properties hold we shall say that the triple (A, R, P ) satisﬁes the standard hypotheses.

10 Note that, by Corollary 2.13, given (i) and (ii), the final property (iii) holds provided that it holds over residue fields, i.e. provided that for each residue field F the natural map F ⊗A EndR(P ) → EndRF (PF ) is surjective, and therefore an isomorphism.

Example 3.1. (i) Let G be a finite group and let X be a finite G-set. Let S be a commutative ring, Then we have the permutation module SX. For an orbit O of G on X × X we have the SG-module endomorphism αO of SX satisfying αO(x) = (y,x)∈O y, and these elements form an S-basis of End (SX). In particular Λ = End (ZX) is free of finite rank over Z. It SG P ZG follows that the ZG-module ZX has the base change property on endomorphism algebras and that (Z, ZG, ZX) satisfies the standard hypotheses.

Example 3.2. Let F be a ﬁeld and A = F [t,t−1]. Let n be a positive integer and R = Hec(n)A,t be the corresponding Hecke algebra of type A. By work of Dipper and James, [4], Theorem 3.3(i), if P is a (quantised) Young permutation module for R, then (A, R, P ) satisﬁes the standard hypotheses. This is a key example for us and we shall give more details, in Section 5, when we have more notation available.

We will assume until the end of this section that the triple (A, R, P ) satisﬁes the standard hypotheses. Applying Corollary 2.13 with Λ = EndR(P ) we get the following. Proposition 3.3. Suppose that for every residue ﬁeld F of A the natural map F ⊗A DEndR(P ) → DEndRF (PF ) is surjective. Then for any A-algebra B the natural map

op B ⊗A DEndR(P ) → DEndRB (PB) is an isomorphism.

Let K be the ﬁeld of fractions of A.

Lemma 3.4. Let F be a residue ﬁeld of A.

(i) We have dimF AnnRF (PF ) ≥ dimK AnnRK (PK ) with equality if and only if the natural map F ⊗A AnnR(P ) → AnnRF (PF ) is an isomorphism.

(ii) We have dimF DEndRF (PF ) ≥ dimK DEndRF (PF ) with equality if and only if the natural map F ⊗ADEndR(P ) → DEndRF (PF ) is an isomorphism.

Proof. (i) Let h = dimK AnnRK (PK ). We have the isomorphism K ⊗A

AnnR(P ) → AnnRK (PK ) so the torsion free rank of AnnR(P ) is h. Now we have the injective map F ⊗A AnnR(P ) → AnnRF (PF ) so that dimF AnnRF (PF ) ≥ h with equality if and only F ⊗AAnnR(P ) → AnnRF (PF ) is an isomorphism. (ii) Similar.

11 Lemma 3.5. Suppose that RK is semisimple. Let F be a residue ﬁeld of A. If PF has the double centraliser property then the natural maps

F ⊗A AnnR(P ) → AnnRF (PF ) and F ⊗A DEndR(P ) → DEndRF (PF ) are isomorphisms.

Proof. Let n be the rank of R as A-module. Since RK is semisimple, PK has the double centraliser property by the Jacobson Density Lemma, see e.g., [15, XVII,§3, Theorem 1]. Hence we have the short exact sequence

0 → AnnRK (PK ) → RK → DEndRK (PK ) → 0 in particular

n = dimK AnnRK (PK )+dimK DEndRK (PK ) (1).

By assumption, we also have the short exact sequence

0 → AnnRF (PF ) → RF → DEndRF (PF ) → 0.

In particular we have

n = dimF AnnRF (PF )+dimF DEndRF (PF ) (2).

But by Lemma 3.4, dimF AnnRF (PF ) ≥ dimK AnnRK (PK ) and dimF DEndRF (PF ) ≥ dimK DEndRK (PK ) so that (1) and (2) imply that dimF AnnRF (PF ) = dimK AnnRK (PK ) and dimF DEndRF (PF ) = dimK DEndRK (PK ). Another application of Lemma 3.4 completes the proof.

We write Fp for the ﬁeld of prime order p.

Proposition 3.6. Suppose that RK is semisimple. If, for each residue field F , the RF -module PK has the double centraliser property, then the R-module P has the double centraliser property. In particular if G is a finite group and X a finite G-set then the permutation module ZX for ZG has the double centraliser property provided that for each prime p the FpG permutation module FpX has the double centraliser property.

Proof. First suppose that A is a discrete valuation ring. Let π be a generator of the maximal ideal. We need to show that for each σ ∈ DEndR(P ) there exists w ∈ R satisfying σ(p)= wp for all p ∈ P . The mapσ ˜ ∈ EndRK (PK ), obtained from σ by extending coeﬃcients, lies in DEndRK (PK ) so there is k somew ˜ ∈ RK such thatσ ˜(p) =wp ˜ , for all p ∈ P . Now we have π w˜ ∈ R for some positive integer k. So we get that, for some w ∈ R, πkσ(p) = wp, for

12 all p ∈ P . We shall say that σ has index k if k is the smallest non-negative integer such that there exists w ∈ R such that πkσ(p)= wp, for all p ∈ P . Suppose that the proposition is false and that σ is an element of DEndR(P ) with smallest possible positive index. Suppose ﬁrst that σ has index 1. Then, for some w ∈ R, we have πσ(p) = wp, for all p ∈ P . Let w be the image of w in RF = R/πR. Then w acts as zero on PF so that w ∈ AnnRF (PF ), which by the hypothesis and Lemma 3.5, is AnnR(P )+πR. ′ ′ Thus we have w = t + πw , for some t ∈ AnnR(P ) and w ∈ R. Hence we have πσ(p) = πw′p, for all p ∈ P , and therefore σ(p) = w′p, for all p ∈ P and σ has index 0, a contradiction. So suppose σ has index k > 1. Then πσ has index k − 1 so, by the minimality of k, there exists w ∈ R such that πσ(p)= wp, for all p ∈ P . But then σ has index 1, and this case has already been ruled out. We now allow A to be an arbitrary Dedekind domain. We consider the natural map ν : R → DEndR(P ). Let m be a maximal ideal of A then we have the localised map νm : Rm → DEndR(P )m = DEndPm (Pm), and this we know to be an epimorphism by the case already considered, since Am is a discrete valuation ring. Hence νm is an epimorphism, for each maximal ideal m and so ν is an epimorphism, by [1, Proposition 3.9].

Example 3.7. We give an example to show that base change does not always holds on annihilators of permutation modules. Let A be a commutative ring. Let G be a ﬁnite group and T a subgroup. We consider the G-set X = G/T . The permutation module AX may be realised as the left ideal of AG generated by t∈T t. Consider a function f : G → A. The element d = f(g)g belongs to the annihilator of the AG-module AX if and only g∈G P if f(g)gzt = 0 for all z ∈ G. Now the coeﬃcient of u ∈ G in P g∈G,t∈T f(g)gzt is f(ut−1z−1), so that the condition for d to belong gP∈G,t∈T t∈T to the annihilator is f(ut−1z−1) = 0 for all u, z ∈ G, or P Pt∈T P f(xty−1) = 0 tX∈T for all x,y ∈ G. Putting H = G × G, we have an action of H on G by h · g = xgy−1, for h = (x,y), g ∈ G. Hence the condition is

f(h · t) = 0 (3) tX∈T for all h ∈ H. We write C(G, T, A) for the A-module of all functions f : G → A satisfying (3). Now suppose that t is an element of order 2 and T = {1,t}. We can make a graph Γ(G, T ) out of this data, with vertex set G and edges {h · 1, h · t}, h ∈ H. Now C(G, T, A) is the A-module of all A-valued functions on G satisfying f(g1)+ f(g2) = 0, whenever {g1, g2} is an edge.

13 We consider an arbitrary finite simple graph Γ = (V, E) and the A-module of functions C(Γ, A) of all functions f : V → A such that f(v)+ f(w) = 0, r whenever {v, w} is an edge. Then C(Γ, A) = i=1 C(Γi, A), where Γi = (V , E ), 1 ≤ i ≤ r, are the connected components of Γ. i i L Let F be a field. If F has characteristic 2 then C(Γi, F ) consists of the constant functions on Vi , so that dimF C(Γi, F ) = 1 and dimF C(Γ, F )= r. However, if the characteristic of F is not 2 then C(Γi, F ) is 0 if Γi contains and odd cycle and otherwise is one dimensional. Hence dimF C(Γ, F ) is the number of components without odd cycles and in particular dimF C(Γ, F )= r if Γ contains no odd cycles and dimF C(Γ, F ) < r if Γ contains an odd cycle. Thus if Γ contains an odd cycle then dimF C(Γ, F ) > dimK C(Γ, K), for a field F of characteristic 2 and a field K of characteristic different from 2. Applying this to Γ(G, T ) we have that if this graph contains an odd cycle then dimF C(G, T, F ) > dimK C(G, T, K), for a field F of characteristic 2 and a field K of characteristic different from 2. In particular if A is a Dedekind domain with characteristic 0 field of fractions K and a residue field F of characteristic 2 then the AG-module AX does not have the base change property on annihilators. One may give concrete examples as follows. Let m be a positive integer divisible by 4. Let G be the symmetric group Sym(m) of degree m and let t be the regular involution (13)(24)(57)(68) . . ., let x = (123)(567)(9, 10, 11) . . . and let y be the inverse of (234)(678)(10, 11, 12) . . .. Let h = (x,y). Then the vertices 1, h · 1= t and h · t = h2 · 1= x2y−2 belong to a cycle of length 3. Hence dimF AnnFG(F X) > dimK AnnKG(KX). Hence by Lemma 3.4, for X = H/T , the permutation module ZX does not have base change on annihilators. Moreover, by Lemma 3.5, the permutation module F2X for F2G does not have the double centraliser property.

Part II: Schur algebras, Hecke algebras and symmetric groups

4 The double centraliser property for some modules for the quantised Schur algebras

We ﬁx a ﬁeld F and 0 6= q ∈ F . In treating quantum groups over F at parameter q, and their representations, we suppress q from the notation whenever possible, in the interests of simplicity.

14 For a positive integer n we write G(n) for the corresponding quantum general linear group, as in [6]. We write X(n) for Zn and X+(n) for the set of elements λ = (λ1, . . . , λn) ∈ X(n) such that λ1 ≥···≥ λn. As in [11], we write Λ(n) for the set of n-tuples of non-negative integers and Λ+(n) for the set of partitions with at most n parts, i.e Λ+(n) = X+(n) Λ(n). By the degree deg(α) of a finite string of non-negative integers α = (α , α , · · · ) we T 1 2 mean the sum α1 +α2+· · · . For r ≥ 0 we write Λ(n, r) for the set of elements of Λ(n) that have degree r and write Λ+(n, r) for X+(n) Λ(n, r). We have the usual (dominance) partial order on Λ(n). Thus α = (α ,...,α ) ≤ β = T 1 n (β1,...,βn) if α and β have the same degree and α1 +· · ·+αi ≤ β1 +· · ·+βi, for all 1 ≤ i ≤ n. We have the Borel subgroup B(n) and maximal torus T (n) as in [6]. Weights of G(n)-module and B(n)-modules (and T (n)-modules) are com- puted with respect to T (n). For each λ ∈ X+(n) we have a simple G(n)- module L(λ) with highest weight λ and the modules L(λ), λ ∈ X+(n), form a complete set of pairwise non-isomorphic simple G(n)-modules. For λ ∈ Λ(n) we have a one dimensional B(n)-module Fλ with weight G(n) + λ, and the induced module indB(n)Fλ, as in [6, 0.21]. For λ ∈ X (n) the G(n) induced module indB(n)Fλ is a non-zero finite dimensional module, which we denote ∇(λ). For λ ∈ Λ+(n) the module ∇(λ) is polynomial of degree deg(λ). Recall that a finite dimensional G(n)-module M is called a tilting module if both M and the dual module module M ∗ admit filtrations with sections of the form ∇(λ), λ ∈ X+(n). For λ ∈ X+(n) there is an indecomposable tilting module T (λ), with highest weight λ, and the modules T (λ), λ ∈ X+(n), form a complete set of indecomposable tilting modules. For λ ∈ Λ+(n), the module T (λ) is polynomial of degree deg(λ). The category of (left) G(n)-module which are polynomial of degree r is naturally equivalent to the category of (left) modules for the (quantised) Schur algebra S(n, r). The Schur algebra S(n, r) has the structure of a quasi- hereditary algebra with respect to the natural partial order on Λ+(n, r) and labelling of simple modules L(λ), with costandard modules ∇(λ), and tilting modules T (λ), λ ∈ Λ+(n, r) (for details see for example [6, Appendix]). We shall say that a subset σ of Λ+(n, r) is saturated (resp. co-saturated) if whenever µ ∈ σ, λ ∈ Λ+(n, r) and λ ≤ µ (resp. λ ≥ µ) then λ ∈ σ. For a finite dimensional algebra S over F a finite dimensional indecomposable module U and a finite dimensional S-module V we write (V | U) for the number of components isomorphic to U in a decomposition of V as a direct sum of indecomposable modules. For a tilting module T polynomial of degree r we set

+ πs(T )= {λ ∈ Λ (n, r) | (T | T (λ)) 6= 0} (the subscript s indicates that the tilting module corresponds to the signed

15 Young modules that appear in Section 5). By [8, Proposition 2.1] we have the following.

Proposition 4.1. If πs(T ) is saturated then T , as an S(n, r)-module, has the double centraliser property.

We write E for the natural G(n)-module. For r ≥ 0 we have the rth (quantised) exterior power r E, as in [6, 1.2] and, for a ﬁnite string of non- negative integers α = (α , α ,...) the polynomial module 1 V2 α E = α1 E ⊗ α2 E ⊗··· . We shall use that if β is a string obtained by reordering the parts of α then β E is isomorphic to α E, see [6, Remark V V V (i) following 3.3 (1)]. V V The transpose of a partition λ will be denoted λ′. For λ ∈ Λ+(n) we have

′ λ E = T (λ) ⊕ C (1) where C is a direct sum of tilingV modules T (µ) with µ smaller than λ in the dominance order. We are interested in conditions that ensure that a direct sum of modules of the form α E has the double centraliser property. Let r ≥ 0. We write Par(r) for the set of partitions of r. For a subset σ of V Par(r) we write σ′ for the set {λ′ | λ ∈ σ}. For λ ∈ Par(r) the module λ E is non-zero if and only if λ ∈ Λ+(n, r)′. Moreover, λ E has highest weight λ V λ′, for λ ∈ Λ+(n, r)′. Now if M = ⊕ ′ ( E)(dλ) and if µ is minimal λ∈Λ(n,r) V such that d 6= 0 then µ′ is a highest weight weight of M and occurs with µ V λ (eλ) multiplicity dµ. It follows that if we can also write M = ⊕λ∈Λ(n,r)′ ( E) µ (dµ) then dµ = eµ. Thus M = N ⊕ ( E) where V

λ V (dλ) λ (eλ) N = ( E) = λ∈Λ(n,r)′\{µ}( E) λ∈Λ(n,r)′\{µ} M V L V + ′ so we get by induction that dλ = eλ, for all λ ∈ Λ (n, r) . So we have: Lemma 4.2. If the ﬁnite dimensional S(n, r)-module T is a direct sum of λ + ′ modules of the form E, λ ∈ Λ (n, r) , then the multiplicities dλ in an expression V T = ( λ E)(dλ) λ∈Λ+(n,r)′ M V are uniquely determined.

λ (dλ) For a module T of the form T = λ∈Λ+(n,r)′ ( E) we write ζs(T ) for the set of λ ∈ Λ+(n, r)′ such that d 6= 0. Lλ V Lemma 4.3. Assume r ≤ n. Let T be a ﬁnite direct sum of modules of λ ′ the form E, λ ∈ Par(r). Then ζs(T ) ⊆ πs(T ), with equality if ζs(T ) is co-saturated. V

16 λ Proof. Let λ ∈ ζs(T ). Then E is a direct summand of T and, by (1), T (λ′) is a direct summand of λ E, and hence of T , so that λ′ ∈ π (T ). V s Now suppose that ζ (T ) is co-saturated. Let λ ∈ π (T ). Then T (λ) is s V s a direct summand of T and hence of some summand µ E of T . Then ′ ′ µ ∈ ζs(T ) and, by (1), we have λ ≤ µ . Hence λ ≥ µ and, since ζs(T ) is ′ ′ V co-saturated, we have λ ∈ ζs(T ), i.e. λ ∈ ζs(T ) .

Note that if T is as in the above lemma with ζs(T ) cosaturated then πs(T ) is saturated and so T , as a module for S(n, r), has the double centraliser property by Proposition 4.1. We can improve on this basic situation somewhat. We recall, [16, I, Section 6], the notion of a reﬁnement of a partition. If λ and µ are partitions then we write λ ∪ µ for the partition made from all the parts of λ and µ by arranging them in descending order. Let µ = (µ1,µ2,...) be a partition. A partition λ is a reﬁnement of µ if it is possible to write (i) (i) λ = ∪i≥1λ , where each λ is partition of µi. In this situation we shall also say that µ is a coarsening of λ. We shall say that µ is a simple coarsening of λ if the parts of µ are λ1, . . . , λi−1, λi + λj, λi+1, . . . , λj−1, λj+1, λj+2,... (arranged in descending order) for some 1 ≤ i < j. Note that µ is coarsening of λ if and only if, for some positive integer m, there exists a sequence of partitions µ = µ(1),µ(2),...,µ(m)= λ, such that µ(k) is a simple coarsening of µ(k + 1), for 1 ≤ k

Proposition 4.4. Assume n ≥ r. Let T be a module of the form λ (dλ) ˆ λ∈Λ+(n,r)( E) , for some non-negative integers dλ. If ζs(T ) is co- saturated then T has the double centraliser property. L V

Proof. We argue by induction on |ζˆs(T )\ζs(T )|. If this is 0 then ζs(T ) is co- saturated and so T is a double centraliser module by the remarks preceding the lemma. Now suppose that h = |ζˆs(T )\ζs(T )| > 0 and that the result λ holds for all direct sums U of modules of the form E, with ζˆs(U) co- saturated and |ζˆ (U)\ζ (U)| < h. s s V We choose µ ∈ ζˆs(T )\ζs(T ). Then there exists λ ∈ ζs(T ) and partitions µ(1),...,µ(m) such that µ = µ(1), λ = µ(m) and µ(k) is a simple coarsening of µ(k + 1), for 1 ≤ k

|ζˆs(U)\ζs(U)| = |ζˆs(T )\ζs(T )|− 1= h − 1.

Hence U has the double centraliser property and so too does T .

Example 4.5. We note the following special case. Suppose that r = a+b ≤ n. We consider the coarsening σ of σ = {(a, 1b)}. We claim that

+ σˆ = {λ = (λ1, λ2,...) ∈ Λ (n, r) | λ1 ≥ a}.

b Let λ ∈ σˆ. Then λ ≥ (a, 1 ) so λ1 ≥ 1. If λ1 = a then (λ2, λ3,...) is a partition of b. Clearly any partition of b is a coarsening of 1b and so λ = b (a, λ2, λ3,...) is a coarsening of (a, 1 ). If λ1 > a then for some i we have λi > a, λi+1 ≤ a. We may assume inductively that µ = (λ1, . . . , λi−1, λi − b−1 1, λi+1,...) is a coarsening of (a, 1 ). Hence µ ∪ {1} is a coarsening of (a, 1b) and λ is a coarsening of µ ∪ {1} and hence of (a, 1b).

Corollary 4.6. Let 1 ≤ r ≤ n and let a1, . . . , am be positive integers and b1, . . . , bm be non-negarive integers such that ai + bi = r for 1 ≤ i ≤ m. Then the tilting module

m T = ai E ⊗ E⊗bi i=1 M V for S(n, r), has the double centraliser property.

b Proof. Let σi = {(ai, 1 i )}. Thenσ ˆi = {λ ∈ Par(r) | λ1 ≥ ai} so that

m m ζˆs(T )= σˆi = {λ ∈ Par(r) | λ1 ≥ ai} i[=1 i[=1 = {λ ∈ Par(r) | λ1 ≥ a} where a is the minimum of a1, . . . , am, by the example, and this is a co- saturated set so T has the double centraliser property, by Proposition 4.4.

Remark 4.7. Let σ be a finite set of finite strings of non-negative integers of fixed degree r. It would be interesting to have a precise combinatorial λ criterion on σ for the S(n, r)-module λ∈σ E, to have the double centraliser property. Some examples in which this fails are given in [8, Section L V 6]. We shall also see, in Section 7 below, that it fails for the S(4, 4)-module 2 E ⊗ 2 E in characteristic 2. V V

18 5 The double centraliser property for some modules for the Hecke algebra

We will be working with quantised Young permutation modules over a Hecke algebra. But ﬁrst we record the following in the general context. It will be used to twist with the sign representation. We leave the details to the reader.

Remark 5.1. Let R be a finite dimensional algebra over a field F and let α be an algebra automorphism of R. For a finite dimensional module V , α affording representation ρ : R → EndF (V ), we denote by V the corresponding twisted module, i.e. the vector space V regarded as an R-module α via the representation ρ ◦ α : R → EndF (V ). The annihilator AnnRV is −1 α equal to α (AnnR(V )). Further DEndR(V ) = DEndR(V ) and V has the double centraliser property if and only if V α has.

We now make a general remark on double centralisers. Let S be a finite dimensional algebra over a field F . Let e be a non-zero idempotent in S. Thus we have the algebra eSe. The argument given below is taken from the proof of [8, Proposition 7.1] (but since the context is slightly different we give it again).

Proposition 5.2. Suppose T is a ﬁnite dimensional module such that the natural map EndS(T ) → EndeSe(eT ) is an isomorphism. If T is a double centraliser module for S then eT is a double centraliser module for eSe.

Proof. Let φ ∈ DEndeSe(eT ). We deﬁne φˆ ∈ EndF (T ) by

φ(x), if x ∈ eT ; φˆ(x)= (0, if x ∈ (1 − e)T.

Let g ∈ EndS(T ) and let f = g | eT ∈ EndeSe(eT ). Then for x ∈ eT , we have

φˆ ◦ g(x)= φ(f(x)) = f(φ(x)) = g ◦ φˆ(x) and, for x ∈ (1 − e)T , we have

φˆ ◦ g(x)= φˆ(g(1 − e)x)= φˆ((1 − e)g(x)) = 0 and gφˆ(x) = 0 (since x ∈ (1−e)T ). Hence we have φˆ◦g = g◦φˆ and therefore φˆ ∈ DEndS(T ). Since T has the double centraliser property for some s ∈ S we have φˆ(x)= ρ(s)(x), for all x ∈ T , where ρ : S → EndF (T ) is the representation aﬀorded by T . Now for x ∈ eT we have φ(x) = eφ(x) = eφˆ(x) = esx = esex, so we have φ = ρe(ese), where ρe : eSe → EndF (eT ) is the representation aﬀorded by the eSe-module eT . Hence eT has the double centraliser property.

19 We shall apply the results of Section 4 to the representation theory of Hecke algebras. We ﬁx a positive integer n. As usual we take A to be a Dedekind domain or a ﬁeld. Let q be a unit in A. We have the Hecke algebra Hec(n)A,q, with the usual generators Ti = Ti,A,q, 1 ≤ i

x(α)= x(α)A,q = Tw,A,q w∈Sym(X α) and N−l(w) y(α)= (−q) Tw,A,q w∈Sym(X α) n of Hec(n)A,q, where N = 2 and where l(w) denotes the length of an element w of Sym(n). We define a subset J(α) of {1,...,n − 1} as follows. We define J1(α) to be the set of i such that 1 ≤ i ≤ α1 and , for r > 1 define Jr(α) to be the set of i such that α1 + · · · + αr−1

20 Dedekind domain or a ﬁeld. Let q = φ(q). Then we have the induced A- ¯ algebra homomorphism Hec(n)A,q → Hec(n)A,¯ q. We identify A⊗A Hec(n)A,q ¯ with Hec(n)A,¯ q via the induced isomorphism A ⊗A Hec(n)A,q → Hec(n)A,¯ q. Thus, if M is a Hec(n)A,q-module then we have the Hec(n)A,¯ q-module MA¯ = A¯ ⊗A M obtained by base change. In particular we have A¯ ⊗A M(λ)A,q = M(λ)A,¯ q, for λ ∈ Par(n). Proposition 5.3. Let M,N be Young permutation modules (resp. signed Young permutation modules) for Hec(n)A,q. Then the natural map

A¯ ⊗ Hom (M,N) → Hom (M ¯,N ¯) A Hec(n)A,q Hec(n)A¯,q A A is an isomorphism.

Proof. We assume that M,N are Young modules (the other case is similar). It is enough to consider the case in which M = M(λ)A,q, N = M(µ)A,q, for some λ, µ ∈ Par(n), and the result holds in this case by the explicit description of spaces of homomorphisms given by Dipper and James in [4, Theorem 3.4]. Thus by Corollary 2.13 we have the following.

Corollary 5.4. Let P be a Young permutation module or a signed permutation module. For any A-algebra B the natural map

op ΨB : B ⊗A EndR(P ) → EndRB (PB) is an isomorphism.

We now focus on the case in which the coefficient ring is a field F and write simply Hec(n) for Hec(n)F,q, write M(λ) for M(λ)F,q and write Ms(λ) for Ms(λ)F,q, λ ∈ Par(n). We shall need the symmetric powers of the natural module for the corresponding quantum general linear group G(n). For a ≥ 0 we write SaE for the ath (quantised) symmetric power of E and for a finite string α = (α1, α2,...) of non-negative integers, write SαE for the tensor product Sα1 E ⊗ Sα2 E ⊗··· , as in [6, Section 2.1]. For λ ∈ Λ+(n) we write I(λ) for the injective envelope of L(λ) in the polynomial category. If λ has degree r then I(λ) is polynomial of degree r. If r ≤ n then for λ ∈ Λ+(n, r) the module SλE is injective, in the polynomial category. We identify Hec(n) with the algebra eS(n,n)e, for a certain idempotent e ∈ S(n,n), as in [6, Section 2.1] (or [11, Chapter 6] in the classical case q = 1). We have the Schur functor f : mod(S(n,n)) → mod(Hec(n)), as in λ λ [11, Chapter 6]. Then we M(λ)= fS E and Ms(λ)= f E, λ ∈ Par(n). We have the module Y (λ)= fI(λ) and Sp(λ)= f∇(λ), the quantised Young V module and Specht module corresponding to λ, for λ ∈ Par(n). We also have λ signed versions Ms(λ)= f E, Ys(λ)= fT (λ), of the Young permutation module and Young module, for λ ∈ Par(λ). (The modules Y (λ), λ ∈ Par(n), V s

21 are, in the classical case, special cases of the modules called signed Young modules in [7]. However the modules Ys(λ) are sufficient for our present purpose.) The modules Y (λ) (resp. Ys(λ)), for λ ∈ Par(n), are indecomposable and pairwise non-isomorphic. By a quantised Young module (resp. signed quantised Young module) we mean a finite direct sum of modules Y (λ) (resp. Ys(λ)), λ ∈ Par(n). By a quantised Young permutation module (resp. signed quantised Young permutation module) we mean a finite direct sum modules M(λ) (resp. Ms(λ)), λ ∈ Par(n). For a Young module U we set π(U)= {λ ∈ Par(n) | (U | Y (λ)) 6= 0}. For a signed Young module V we set πs(V )= {λ ∈ Par(n) | (V | Ys(λ)) 6= 0}. Remark 5.5. We shall need to use the fact that if T,U are tilting modules for S(n,n) then the Schur functor induces an isomorphism HomS(n,n)(T,U) → HomHec(n)(fT, fU). It suffices to observe this in the case in which U = T (λ), for some λ ∈ Par(n). Moreover, since T (λ) is a ′ ′ direct summand of λ E, it is enough to note this in the case U = λ E, and this is true by [6, 2.1,(16)((iii)]. V V Our main interest is in the study of double centralisers and annihilator ideals of Young permutation modules and signed Young permutation modules. But for completeness we also consider the situation for Young modules and signed Young modules. ♯ By [6, 4.4, (10)] and [6, 2.1,(20)] we have M(λ) = Ms(λ), for λ ∈ Par(n). A similar relationship exists between, the Young modules and their signed analogues. This is not recorded in [6]. However the result is

′ ♯ Ys(λ)= Y (λ ) (1) for λ ∈ Par(n), and this follows from the argument of [5, (3.6) Lemma (ii)], applied in the set-up of [6, Section 4.4]. We now consider the double centraliser property for (quantised) Young permutation modules and their signed versions. For λ ∈ Par(n) we may (aλµ) write M(λ) = µ∈Par(n) Y (µ) where aλµ = (M(λ) | Y (µ)). More- over, we have a = 1 and λ ≥ µ whenever a 6= 0. Thus the matrix λλL λµ (aλµ)λ,µ∈Par(n) is invertible and this gives the following for Young permutation modules in the case in which A is a ﬁeld. The result then follows in general by considering MF , where F is a residue ﬁeld of A. The version for signed Young permutation modules may be obtained in a similar fashion.

Lemma 5.6. Let A be a Dedekind domain or a ﬁeld and q a unit in A. If M is a Young permutation module (resp. signed permutation module) then the (dλ) multiplicities dλ (resp. eλ), λ ∈ Par(n), in an expression M = λ M(λ)A,q (eλ) (resp. M = λ Ms(λ)A,q ) are uniquely determined. L L 22 Let A be a Dedekind domain or a ﬁeld and q a unit in A. For a Young (dλ) permutation module M = λ M(λ)A,q we set ζ(M) = {λ ∈ Par(n) | dλ 6= (eλ) 0}. For a signed Young permutationL module N = λ Ms(λ)A,q we set ζ (N)= {λ ∈ Par(n) | e 6= 0}. s λ L Note that this notation is consistent with the earlier usage in the sense that if T is tilting module for S(n,n) then πs(T ) = πs(fT ) and if T is a tilting module of exterior type, for S(n,n) then ζs(T )= ζs(fT ). Proposition 5.7. Let V be a Young permutation module (resp. signed Young permutation module) and suppose that ζˆ(V ) (resp. ζˆs(V )) is co-saturated. Then V has the double centraliser property. Proof. We consider the signed version. (The other case follows from Remark 5.1). Let σ = ζs(V ). We may assume that V = λ∈σ Ms(λ), by [8, (1.5)]. Consider the S(n,n)-module L T = λ E. λ∈σ M V Sinceσ ˆ is co-saturated the S(n.n)-module T has the double centraliser property, by Proposition 4.4. Applying the Schur functor we obtain that V has the double centraliser property, by Proposition 5.2.

Corollary 5.8. Let V be a Young module (resp. signed Young module) and suppose that π(V ) (resp. πs(V )) is co-saturated. Then V has the double centraliser property. Proof. We prove the version for Young modules. By [8, (1.5)] we may assume that V = λ∈π(V ) Y (λ). Consider the Young permutation module U = M(λ). Since Y (λ) is a direct summand of M(λ), for λ ∈ Par(n), λ∈π(V ) L the module V is a direct summand of U. Moreover, an indecomposable L summand of M(λ) has the form Y (µ), for some µ ≥ λ and, since π(V ) is co- saturated, each summand of U has the form Y (µ) for some µ ∈ π(V ). Hence V and U have the same indecomposable summands (up to isomorphism) and hence V has the double centraliser property if and only if U has, by [8, (1.5)]. But U has the double centraliser property by the Proposition, so we are done.

We now turn our attention to the integral case. Let A be a Dedekind domain or ﬁeld and let q be a unit in A. Let K be the ﬁeld of fractions of A. Recall that the Hecke algebra Hec(n)K,q is semisimple if either q is not a root of unity or if q = 1 and K has characteristic 0. For λ ∈ Par(n) we write dim(λ) for the dimension of the Specht module Sp(λ). Thus dim(λ) is the dimension of the ordinary irreducible module corresponding to λ for the symmetric group and the for the Hecke algebra at non-zero parameter which is not a root of unity.

23 Proposition 5.9. Let A be a Dedekind domain or a ﬁeld and let q be a unit in A. Let P be a Hec(n)A,q-module. Suppose that either P is a Young permutation module and that ζˆ(P ) is co-saturated or P is a signed Young permutation and ζˆs(P ) is co-saturated. We write σ for ζ(P ) if P is a Young permutation module and for ζs(P ) if P is a signed Young permutation module. (i) If A = F is a ﬁeld then

2 dimF AnnHec(n)F,q (P )= n! − dim(λ) λX∈σˆ and 2 dimF DEndHec(n)F,q (P )= dim(λ) . λX∈σˆ (ii) If F is a residue ﬁeld of A and q is the image of q in F then the natural maps

F ⊗A AnnHec(n)A,q (P ) → AnnHec(n)F,q (PF ) and

F ⊗A DEndHec(n)A,q (P ) → DEndHec(n)F,q (PF ) are isomorphisms.

Proof. We treat the case of Young permutation modules. The signed case is similar. (i) Consider the Hecke algebra Hec(n)F [t,t−1],t. We write (dλ) ˜ P = λ∈Par(n) M(λ)F,q , for certain non-negative integers dλ. We define P (d ) − λ to beL the Hec(n)F [t,t 1],t-module λ∈Par(n) M(λ)F [t,t−1],q. −1 We regard F as a reside field ofL the ring of Laurent polynomials F [t,t ] with corresponding ideal generated by t−q. Thus we have P˜F = P . The field −1 of fractions of F [t,t ] is the field of rational functions F (t) and Hec(n)F (t),t is semisimple. Moreover, P has the double centraliser property, by Propo- sition 5.7, so that, by Lemma 3.5, the natural map

− ˜ F ⊗ 1 Ann − (P ) → Ann (P ) (1) F [t,t ] Hec(n)F [t,t 1],t Hec(n)F,q is an isomorphism. Now the irreducible constituents of the semisimple ˜ Hec(n)F (t),t-module PF (t) are the Specht modules Sp(λ)F (t),t, with λ ∈ σˆ. Hence

˜ 2 dimF (t) AnnHec(n)F (t),t (PF (t)) = dimF (t) Hec(n)F (t),t − dim(λ) λX∈σˆ = n! − dim(λ)2. λX∈σˆ

24 ˜ 2 Hence the rank of Ann − (P ) is n! − dim(λ) and by (1) this Hec(n)F [t,t 1] λ∈σˆ is also the F -dimension of Ann (P ), giving the ﬁrst assertion. The Hec(n)F,q P second assertion follows from the fact that we have the short exact sequence

0 → AnnHec(n)F,q (P ) → Hec(n)F,q → DEndHec(n)F,q (P ) → 0.

(ii) Let K be the ﬁeld of fractions of A. By part (i) we have

2 dimK AnnHec(n)K,q (PK ) = dimF AnnHec(n)F,q (PF )= n! − dim(λ) λX∈σˆ and

2 dimK DEndHec(n)K,q (PK ) = dimF DEndHec(n)F,q (PF )= dim(λ) . λX∈σˆ

Hence the torsion free rank of AnnHec(n)A,q (P ), as an A-module, is equal to dimF AnnHec(n)F,q (PF ) and the torsion free rank of DEndHec(n)A,q (P ) is equal to dimF DEndHec(n)F,q (PF ). Hence we have

dimF F ⊗A AnnHec(n)A,q (P ) = dimF AnnHec(n)F,q (PF ) and

dimF F ⊗A DEndHec(n)A,q (P ) = dimF DEndHec(n)F,q (PF ) so the injective maps

F ⊗A AnnHec(n)A,q (P ) → AnnHec(n)F,q (PF ) and

F ⊗A DEndHec(n)A,q (P ) → DEndHec(n)F,q (PF ) are isomorphisms.

Now from Proposition 3.3 we get the following.

Corollary 5.10. Let A be a Dedekind domain or a ﬁeld and let q be a unit in A. Let P be a Hec(n)A,q-module. Suppose that either P is a Young permutation module and that ζˆ(P ) is co-saturated or P is a signed Young permutation and ζˆs(P ) is co-saturated. Then P has the double centraliser property and for every A-algebra B the natural map

op B ⊗A DEndHec(n)A,q (P ) → DEndB⊗AHec(n)A,q (B ⊗A P ) is an isomorphism.

25 6 Example: Partition Algebras

In the examples we shall only treat the case of Young modules and leave the analogous case of signed Young modules to the interested reader. We fix a positive integer n. Let A be a Dedekind domain or a field and q a unit in A. By a hook permutation module for Hec(n)A,q we mean a b non-zero finite direct sum of modules of form M(a, 1 )A,q, with a + b = n. Definition 6.1. We define the index index(H) of a hook permutation module H for Hec(n)A,q to be the smallest a ≥ 1 such that H, written as a direct b sum of modules of the form M(λ)A,q, with λ ∈ Par(n), the module M(a, 1 ) appears as a direct summand of H.

Deﬁnition 6.2. For 1 ≤ m ≤ n we set

2 Nn,m = dim(λ) Xλ where the sum is over all partitions λ of n with ﬁrst part at least m.

We note that if H is a hook permutation module then ζˆ(H) is co-saturated and indeed ζˆ(H) = {λ = (λ1, λ2,...) ∈ Par(n) | λ1 ≥ index(H)}, by Exam- ple 4.5. From Proposition 5.9, Corollary 5.10 and Corollary 2.13 we have the following.

Lemma 6.3. (i) Let F be a ﬁeld and 0 6= q ∈ F . Let H be a hook permutation module for Hec(n)F,q. Then H has the double centraliser property. Furthermore we have

dimF AnnHec(n)F,q (H)= n! − Nn,m and

dimF DEndHec(n)F,q (H)= Nn,m where m = index(H). (ii) Let A be a Dedekind domain or a ﬁeld and let q be a unit in A. Let H be a hook permutation module for Hec(n)A,q. Let B be an A-algebra. Then HB is a double centraliser module for B ⊗A Hec(n)A,q and indeed the natural maps

B ⊗A AnnHec(n)A,q (H) → AnnB⊗AHec(n)A,q (HB) and op B ⊗A DEndHec(n)A,q (H) → DEndB⊗AHec(n)A,q (HB) are isomorphisms.

26 We specialise to the symmetric group case. Let X be a Sym(n)-set. We say that X is a Young Sym(n)-set if it is finite and each point stabilizer is a Young subgroup. For a Young Sym(n)-set X we write ζ(X) for the set of λ ∈ Par(n) such that Sym(λ) is a point stabiliser. We shall say that X is hook Sym(n)-set if ζ(X) consists of partitions of the form (a, 1b) (with a + b = n). We define the index index(X) of a hook Sym(n)-set to be the smallest integer a ≥ 1 such that the Young subgroup Sym(a, 1b) is a point stabilizer. We shall give an application to partition algebras. Let V be a free abelian group of rank n on basis v1, . . . , vn. For a commutative ring B we have the B- module VB = B⊗Z V and this is a free B-module with basis 1⊗v1,..., 1⊗vn. The symmetric group Sym(n) acts on VB, by permuting the basis elements, ⊗r ⊗r and so on the r-fold tensor product VB = VB ⊗B ···⊗B VB. Thus VB is a permutation module for BSym(n), and hence also for BSym(m), for 1 ≤ m ≤ n (where Sym(m) is identified with the subgroup of Sym(n) consisting of the elements that fix m + 1,...,n). Let I(n, r) be the set of maps from {1, . . . , r} to {1,...,n}. We regard i as an r-tuple (i1,...,ir) with entries in {1,...,n} (where ik = i(k), 1 ≤ k ≤ r). The elements ⊗r vi, i ∈ I(n, r), form a Z-basis for V and wvi = vw◦i, for w ∈ Sym(n), ⊗r ⊗ ⊗r i ∈ I(n, r). We identify VB with VB in the obvious way. Thus VB is the BSym(n) permutation module on the Sym(n)-set I(n, r). Now the point stabilizer of i ∈ I(n, r) in Sym(n), is the subgroup consisting of elements constant on the image of i. Hence the stabilizer in Sym(n) is conjugate to Sym(a, 1b), where b = |Im(i)|. Thus

ζ(I(n, r)) = {(a, 1b) ∈ Par(n) | 1 ≤ b ≤ r}.

Now suppose that m ≤ n. Then w ∈ Sym(m) ﬁxes i if an only if it ﬁxes all elements of Im(i) {1,...,m}. Thus, for m ≤ n, I(n, r) is a hook Sym(m)- set and: T

{(a, 1b) ∈ Par(m) | 1 ≤ b ≤ r}, if m = n; ζ(I(n, r)|Sym(m))= (1). ({(a, 1b) ∈ Par(m) | 0 ≤ b ≤ r}, if m

index(I(n, r)|Sym(m))= m − min(r, m) (2). (We are writing min(d, e) for the minimum of non-negative integers d and e.) Hence we also have the following. Proposition 6.4. Let m ≤ n and r ≥ 1. Then the BSym(m)-module V ⊗r has the double centraliser property moreover the natural maps

⊗r ⊗r B ⊗Z AnnZSym(m)(V ) → AnnBSym(m)(VB )

27 and ⊗r ⊗r B ⊗Z DEndZSym(m)(VZ ) → DEndBSym(m)(VB ) ⊗r are isomorphisms. As a B-module AnnBSym(m)(VB ) is free of rank m! − ⊗r Nm,d and DEndB(VB ) is free of rank Md,m, where d = min(m, r).

Recall that, for an element δ of B we have the partition algebra Pr(δ)B over B. One may ﬁnd a detailed account of the construction and properties of Pr(δ)B in for example the papers by Paul P. Martin, [17], [18], and [13], [3]. One also has the half partition algebra Pr+1/2(δ)B , a sub-algebra of ⊗r Pr+1(δ)B . Furthermore Pr(n) and Pr+1/2(n) act naturally on V and these actions commute with the actions of Sym(n) and Sym(n − 1). By the result of Halverson-Ram, [13, Theorem 3.6] the maps

⊗r Pr(n) → EndBSym(n)(V ) (3) and

⊗r Pr+1/2(n) → EndSym(n−1)(V ) (4) given by the representations, are surjective. Thus specialising to m = n or n − 1 in the above proposition we have the following, which is the main result of [3]

Corollary 6.5. For any commutative ring B and positive integers n and r, the natural maps ⊗r BSym(n) → EndPr(n)(V ) and ⊗r BSym(n − 1) → EndPr+1/2(n)(V ) are surjective.

7 Further Examples

We restrict to the classical case q = 1. We fix a discrete valuation ring R with field of fractions K of characteristic 0 and a residue field F of positive characteristic p. Here “base change” will mean base change from R to F . We shall say that base change holds for a finite G-set X, where G is a finite group, if it holds on annihilators i.e. if the natural map F ⊗R AnnRG(RX) → AnnFG(F X) is an isomorphism. We give an example to show that base change and the double centraliser property may fail for a Young permutation module for a symmetric group. Let n be a positive integer. For λ ∈ Par(n) we write simply M(λ) for M(λ)F and Sp(λ) for Sp(λ)F . We use James’s notation, [14], for the irreducible F Sym(n)-modules. A partition λ = (λ1, λ2,...) is called p-regular if no non-zero part is repeated

28 p-times. We write Par(n)reg for the set of p-regular partitions of n. For λ ∈ Par(n)reg the Specht module Sp(λ) has a unique irreducible quotient λ λ D , and the modules D , λ ∈ Par(n)reg, form a complete set of pairwise non-isomorphic irreducible F Sym(n)-modules. For σ ⊆ Par(n) we write C(σ) for the cosaturation, i.e. for the set of µ ∈ Par(n) such that µ ≥ λ for some λ ∈ σ. We fix a Young Sym(n)-set X, i.e. a finite Sym(n)-set such that each point stabiliser is a Young subgroup. We write ζ(X) for the set of λ ∈ Par(n) such that Sym(n) is the stabiliser of some element of X. Now for λ ∈ Par(n), the module Sp(λ)K appears as a composition factor of KX if and only if it appears as composition factor of M(λ)K for some λ ∈ ζ(X). However the composition factors of M(λ)K are precisely those Sp(µ)K with µ ≥ λ, see e.g. [14, 14.1]. Hence the composition factors of KX are precisely those Sp(µ)K with µ ∈ C(ζ(X)). The image of the representation ρ : KG → 2 EndK (KX) has dimension λ∈C(ζ(X))) dim(λ) and the annihilator of KX has dimension n! − dim(λ)2. We define the Sym(n)-set λ∈C(ζ(PX)) P C(X)= Sym(n)/Sym(λ). λ∈Ca(ζ(X)) The above applies to KC(X) so that the annihilator KC(X) also has di- 2 mension n! − λ∈C(ζ(X)) dim(λ) . However, we know that the base change property on ideals holds for C(X) so that AnnF Sym(n)(FC(X)) also has P 2 dimension n! − λ∈C(ζ(X)) dim(λ) . Now AnnF Sym(n)(F X) is the ideal of elements a ∈ F G which act as zero on all M(λ), λ ∈ ζ(X). Hence we have P the following.

Proposition 7.1. Let n be a non-negative integer and let X be a Young Sym(n)-set. Then AnnF Sym(n)(FC(X)) ≤ AnnF Sym(n)(F X) and X has the base change property if and only if AnnF Sym(n)(FC(X)) = AnnF Sym(n)(F X) Remark 7.2. Let Λ be a finite dimensional algebra over a field L. Let M be a finite dimensional Λ-module. We will say that a Λ-module N is a generalised section of M if N is a section of a finite direct sum of copies of M. Let I be the annihilator of M. Then clearly I annihilates any generalised section of M. Suppose N is a finite dimensional Λ-module annihilated by I. The representation ρ : Λ → EndL(M) gives rise to an embedding of Λ/I into a finite direct sum of copies of M. Moreover, there is an epimorphism from a finite direct sum of copies of Λ/I to N and so N is a generalised section of M. Hence N is annihilated by I if and only if it is a generalised section of M. Let J be the Jacobson radical of Λ. If J kM = 0, for some positive integer k, then we have J kN = 0, for any generalised section of N of M. Hence if I annihilates N then the Loewy length of N is less than or equal to the Loewy length of M.

29 Proposition 7.3. Let X be a Young permutation module for the symmetric group Sym(n). Then X has the base change property on ideals if and only if, for all µ ∈ C(ζ(X)), the Young module Y (µ) is a generalised section of F X.

Proof. We form C(X) = λ∈C(ζ(X)) Sym(n)/Sym(λ) as above. Then, by Proposition 7.1, X has the base change property if and only if ` AnnF Sym(n)(F X) = AnnF Sym(n)(FC(X)). Suppose µ ∈ C(ζ(X)). The Young module Y (µ) is a direct summand of M(µ) and hence annihilated by AnnF Sym(n)(FC(X)) and hence annihilated by AnnF Sym(n)(F X). Hence Y (µ) is a generalised section of F X. Now suppose that each Y (µ), with µ ∈ C(ζ(X)) is a generalised section of F X. Then AnnF Sym(n)(F X) annihilates every such Y (µ). However, FC(X) is a direct sum of Young modules Y (µ) with µ ∈ C(ζ(X)). Hence AnnF Sym(n)(F X) annihilates FC(X), i.e. AnnF Sym(n)(F X) ≤ AnnF Sym(n)(FC(X)). The converse is certainly true so AnnF Sym(n)(F X) = AnnF Sym(n)(FC(X)). Hence the dimension of AnnF Sym(n)(F X) is the dimension of AnnF Sym(n)(FC(X)) and, since C(X) has the base change property, this is the K-dimension of AnnKSym(n)(KC(X)), which is also the K-dimension of AnnF Sym(n)(KX) so we are done.

Corollary 7.4. Let X be a Young Sym(n)-set. (i) If Y is also a Young Sym(n) set such that ζ(X)= ζ(Y ) then X has base change if and only if Y has base change. (ii) If base change fails for X then there exists a minimal element λ of ζ(X) such that base change fails for Sym(n)/Sym(λ).

We now put these properties to use in the example that we now give. Since we have shown that base change and the double centraliser property holds for permutation module M(λ) with λ a hook the ﬁrst case to look at here is λ = (2, 2). If the characteristic of F is at least 5 then the semisimplicity of of M(λ) guarantees that the desired properties hold. Suppose now that the characteristic is 3. Then M(2, 2) has image M(4) so that M(2, 2) had the desired property if and only if M(2, 2) ⊕ M(4) has. Moreover M(2, 2) has a ﬁltration 0 = V0 < V1 < V2 < V3 = M(2, 2) with

M(2, 2)/V1 = Sp(3, 1) ⊕ Sp(4) = M(3, 1)

(e.g. by block considerations) so that M(3, 1) is an of M(2, 2) image and M(2, 2) has the desired properties since M(2, 2) ⊕ M(3, 1) ⊕ M(4) has, by [8, (1.4)]. Now suppose F has characteristic 2. We shall use the Schur algebra S(4, 4) and the Schur functor f : mod(S(4, 4)) → mod(F Sym(4)), as in [11,

30 Chapter 6]. Now the injective module I(2, 2) has a filtration with sections ∇(2, 2), ∇(3, 1), ∇(4). The module I(2, 2) is a direct summand of the module S2E ⊗ S2E and this also has a filtration with sections ∇(2, 2), ∇(3, 1), ∇(4). Hence we have I(2, 2) = S2E ⊗ S2E and, applying the Schur functor, the module M(2, 2) = Y (2, 2) has a filtration 0 = V0 < V1 < V2 < V3 = M(2, 2) with V1 = Sp(2, 2), V2/V1 = Sp(3, 1) and V3/V2 = Sp(4). Now F Sym(4) has simple modules D4 = Sp(4) and D3,1 = Sp(2, 2). Moreover the module Sp(3, 1) has a submodule U with Sp(3, 1)/U = D3,1 and U = D4. The simple modules are self dual. The module M(2, 2) is self dual so that D3,1 appears in the head as well as the socle. Moreover, since M(2, 2) is indecomposable the radical of M(2, 2) contains a copy of D3,1. Similarly since D4 appears in the head it also appears in the socle and indeed in the radical of M(2, 2). If follows that M(2, 2) has a submodule Z with both Z and M(2, 2)/Z isomorphic to D3,1 ⊕ D4. Hence M(2, 2) has Loewy length 2. Now consider the module M(3, 1). The module I(3, 1) has sections ∇(3, 1), ∇(4) as does the injective module S3E⊗E. Hence we have I(3, 1) = S3E⊗E and applying the Schur functor we have that M(3, 1) = Y (3, 1) has a filtration with sections Sp(3, 1), Sp(4). Since M(3, 1) is indecomposable it must be uniserial with sections D4, D3,1, D4. This module has Loewy length 3 and so, by Remark 7.2, M(3, 1) can not be a generalised section of M(2, 2). Hence the annihilator of M(2, 2) does not annihilate Y (3, 1). Hence, by Proposition 7.3, Sym(4)/Sym(2, 2) does not have the base change property. In summary, have shown that, taking R a discrete valuation ring with residue field F of characteristic 2, then the Young Sym(4)-set Sym(4)/Sym(2, 2) does not have the base change property, from R to F , on annihilators. By Lemma 3.5, the Young permutation module M(2, 2) does not have the double centraliser property.

8 Annihilators and cell ideals

We make some remarks on the annihilator ideals of Young permutation modules in relation to cell structure, in the spirit of [12], [9], [2]. First consider a semisimple algebra B, ﬁnite dimensional over a ﬁeld F . Let S(φ), φ ∈ Φ, be a complete set of pairwise non-isomorphic irreducible left B-modules. For a subset τ of Φ we write I(τ) for the sum of all irreducible submodules, of the left regular module B, isomorphic to S(φ) for some φ ∈ Φ. Then I(τ) is an ideal of B. We have B = I(τ) ⊕ I(σ), where σ is the complement of τ in Φ and moreover:

I(τ) = AnnB(V ) for any left B-module V of the form (dφ) φ∈Φ S(φ) with dφ 6= 0 if and only if φ ∈ σ. (1) We recallL the notion of a cellular algebra due to Graham and Lehrer, [10].

31 Deﬁnition 8.1. Let A be an algebra over a commutative ring R. A cell datum 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) and an injective map C : λ∈Λ N(λ) × N(λ) → A with image an R-basis of A. (C2) For λ ∈ Λ and u, v ∈ N(λ) we write C(u, v)= Cλ ∈ R. Then ∗ is an ` u,v λ ∗ λ R-linear involutory anti-automorphism of A such that (Cu,v) = Cv,u. (C3) If λ ∈ Λ and u, v ∈ N(λ) then for any element a ∈ A we have

λ ′ λ aCu,v ≡ ra(u , u)Cu′,v (mod A(< λ)) ′ u ∈XN(λ) ′ where ra(u , u) ∈ R is independent of v and where A(< λ) is the R-submodule µ ′′ ′′ of A generated by {Cu′′,v′′ | µ ∈ Λ, µ < λ and u , v ∈ N(µ)}. We shall assume throughout that Λ is ﬁnite. As in [10], for λ ∈ Λ, we write W (λ) for the corresponding cell module. For τ ⊆ Λ we write A(τ) for λ the R- span of all Cu,v with λ ∈ τ and u, v ∈ N(λ). Then A/A(τ) is a R-free λ on {Cu,v + A(τ) | λ ∈ Λ\τ, u, v ∈ N(λ)}. We shall write WR(λ) for W (λ) and AR(τ) for A(τ) if we wish to emphasise the role of the base ring R. Given a homomorphism of commutative rings R → R¯ we obtain, by ¯ base change, a cell structure on AR¯ = R ⊗R A, which will also be denoted (Λ,N,C, ∗), see [10, (1.8)]. For τ ⊆ Λ and λ ∈ Λ, we identify ¯ ¯ A(τ)R¯ = R ⊗R A(τ) with AR¯(τ) with W (λ)R¯ = R ⊗R W (λ) with WR¯(λ) in the natural way. Suppose for the moment that R is a domain with ﬁeld of fractions F . For an R-submodule X we write XF for F ⊗R X and if X is a submodule of A identify XF with a submodule of AF . Then we have

A(τ)= A A(τ)F (2). \ To see this write X = A A(τ)F , the set of all a ∈ A such that ra ∈ A(τ) for some 0 6= r ∈ R. Thus X/A(τ) is R-torsion and since A/A(τ) is torsion T free we have X/A(τ) = 0, i.e. X = A(τ). We assume again that R is an arbitrary commutative ring. If τ is a saturated subset of Λ then A(τ) is an ideal of A, called the cell ideal corresponding to τ. Suppose now that R = F is a field and that A is semisimple. The cell modules W (λ), λ ∈ Λ, form a complete set of pairwise non-isomorphic irreducible A-modules, see [10, (3.8) Theorem]. If τ is a saturated subset of Λ with complement σ then A(τ), as a left A-module has a filtration with quotients the cell modules W (λ), λ ∈ τ, and the quotient A/A(τ) has a filtration with sections the cell modules A(λ), λ ∈ σ. Thus from (1) we have:

32 A(τ)= I(τ) and I(τ) = AnnA(V ), where V is any left module of the (dλ) form λ∈Λ W (λ) with dλ 6= 0 if and only if λ ∈ σ. (3) L Lemma 8.2. Let R be a domain with ﬁeld of fractions F and A an R- ∗ algebra with cell datum (Λ,N,C, ). Suppose that AF is semisimple. Let X be a left A-module, ﬁnitely generated and torsion free over R. Let σ be a (d ) cosaturated subset of Λ. If XF = ⊕λ∈σWF (λ) λ , for positive integers dλ, then AnnA(X) is the cell ideal A(τ), where τ is the complement of σ in Λ.

Proof. For the ideal I(τ) of AF , as deﬁned above we have AnnAF (XF ) = I(τ), by (1), and hence

AnnA(X)= A AnnAF (XF )= A AF (τ)= A(τ) \ \ by (2).

We are interested in taking for A the Hecke algebra Hec(n)R,q, where R is a commutative ring and q a unit in R. Given a homomorphism of commutative ¯ ¯ rings θ : R → R we identify R ⊗R Hec(n)R,q with Hec(n)R,¯ q, where q = θ(q). In particular we may take for the coefficient ring J = Z[x,x−1], the ring of Laurent polynomials with integer coefficients in the indeterminate x, which has quotient field K = Q(x), the field of rational functions. Then Hec(n)K,x is semisimple and the Specht modules Sp(λ)K,x, λ ∈ Par(n), form a complete set of pairwise non-isomorphic irreducible modules. Thus we may take Λ = Par(n) in a cell structure. ∗ We shall say that a cell structure (Λ,N,C, ) on Hec(n)J,x is regular if: (i) Λ = Par(n), with partial order E, the dominance partial order; and (ii) for λ ∈ Par(n) the corresponding cell module W (λ) for the cell structure on Hec(n)J,x is isomorphic to Sp(λ)J,x. We shall say that a cell structure on Hec(n)R,q (for a commutative ring R and unit q of R) is regular if it is obtained by base change from a regular cell structure on Hec(n)J,x. One has available two well known regular cell structures: one coming from the Kazhdan-Lusztig basis on Hec(n)R,q, as in [10], and the other from the Murphy basis, as in [19]. We note that some adjustments to the results that we will refer to from [19], [9], [2] are necessary because the statements there are expressed in terms of right modules, and we are working primarily with left modules. We leave it to the reader to make the appropriate modifications. More importantly we note that in the definition of cell datum used in [9], [2] the ordering on Λ given in condition [10, (1.1) Definition, (C3)] has been reversed. Thus we say that a quadruple (Λ, N,˜ C,˜ ∗) is a datum of the second kind for an algebra A over a commutative ring R if it satisfies conditions (C1), (C2) together with the following.

33 (C˜3) If λ ∈ Λ and u, v ∈ N˜(λ) then for any element a ∈ A we have

˜λ ′ ˜λ aCu,v ≡ ra(u , u)Cu′,v (mod A(> λ)) ′ u ∈XN˜(λ) ′ where ra(u , u) ∈ R is independent of v and where A˜(> λ) is the R- ˜µ ′′ ′′ ˜ submodule of A generated by {Cu′′,v′′ | µ ∈ Λ, µ > λ and u , v ∈ N(µ)}. Given a cell structure of the second kind on an algebra A over a commuta- ˜ ˜λ tive ring R and a subset σ of Λ we write A(σ) for the R-span of all Cu,v with λ ∈ σ, u, v ∈ N˜(λ). If σ is co-saturated then A˜(σ) is an ideal, called a cell ideal. For λ ∈ Λ and v ∈ N˜ (λ) the submodule of A˜(≥ λ)/A˜(> λ) spanned by ˜λ ˜ ˜ ˜ all Cu,v, u ∈ N(λ) has structure independent of v. (Here A(≥ λ)= A(> λ), where ρ = {µ ∈ Par(n) | µ ≥ λ}.) Such a module is denoted W˜ (λ) and called the cell module corresponding to λ. We prefer slightly the outcome of the discussion below when expressed in the original formulation, [10]. So we need to make some minor adjustments in passing between these kinds of cell structure. Let R be a commutative ring and q a unit. We shall consider cell structure ∗ † on the Hecke algebra Hec(n)R,q. We have three R-linear involutions , and ♯ ∗ † of the R-algebra Hec(n)R,q. The ﬁrst two , are anti-automorphisms and ♯ ∗ the third is an automorphism. They are given on generators by Ti = Ti † ♯ and Ti = Ti = −Ti + (q − 1)1, 1 ≤ i

34 the second kind) for Hec(n)R,q (where Par(n) is regarded as a poset via the dominance partial order). The cell module W˜ (λ)= S˜λ is isomorphic to the ∗ λ ∗ dual Specht module, i.e. W˜ (λ) = (S˜ ) is the Specht module Sp(λ)R,q, by [19, Theorem 5.2 and Theorem 5.3]. (In Murphy’s paper the base ring is a domain, but the arguments are valid over any commutative ring - alterna- tively one could appeal to Murphy’s results directly for R = J, q = x and apply base change.) We now consider the cell structure obtained by applying the automorphism ♯. For λ ∈ Par(n) we define N(λ) = N˜ (λ′). For λ ∈ Par(n), λ ♯ s,t ∈ N(λ) we define Cst = xst. Defining C : λ∈Par(n) N(λ) × N(λ) → A = λ Hec(n)R,q by C(s,t)= Cst, for s,t ∈ N(λ), we` have by “transport of struc- ∗ ture”, that (Par(n),N,C, ) is a cell structure on A = Hec(n)R,q. Moreover, ′ for λ ∈ Par(n), we have the cell module W (λ) = W˜ (λ′)♯ = (S˜λ )♯, which, by [19, Theorem 5.2 and Theorem 5.3] is Sp(λ)R,q. Clearly, the cell structure just given is obtained, by base change, from cell structure on Hec(n)J,x constructed in the same way. Hence the structure given on Hec(n)R,q is a regular cell structure. From the definitions we have A˜(σ)♯ = A(σ′), for a subset σ of Par(n) (where σ′ = {λ′ | λ ∈ σ}). We make Lemma 8.2 more explicit in the Hecke algebra case. Lemma 8.3. Let R be a domain, q a unit and F the field of fractions of R. We regard Hec(n)R,q as a cellular algebra via a regular cell datum. Suppose that Hec(n)F,q is semisimple. Let X be a left Hec(n)R,q-module, finitely generated and torsion free over R. Let σ be a cosaturated subset of Par(n). (dλ) If XF = ⊕λ∈σSp(λ)F,q , for positive integers dλ, then

AnnHec(n)R,q (X) = Hec(n)R,q(τ) where τ is the complement of σ in Par(n).

We specialise further to the case in which the module X as above is a permutation module. For a subset σ of Par(n) we write σD for the set of µ ∈ Par(n) such that µ D λ for some λ ∈ σ.

Lemma 8.4. Let R be a domain, q a unit and F the ﬁeld of fractions of R. We regard Hec(n)R,q as a cellular algebra via a regular cell datum. Suppose that Hec(n)F,q is semisimple. Let X be a left Hec(n)R,q-module of the form (dλ) λ∈σ M(λ)R,q for a subset σ of Par(n) with dλ 6= 0 for all λ ∈ σ. Then L AnnHec(n)R,q (X) = Hec(n)R,q(τ) where τ is the complement of σD.

Proof. By Young’s rule, [14, 14.1], the composition factors of XF are Sp(λ)F,q, λ ∈ σD. So the result follows from the previous lemma.

35 Remark 8.5. The main results of [9] are special cases with σ = {λ} for some λ ∈ Par(n). We get [9, Theorem 5.2] by taking R = F in the above lemma and we get [9, Theorem 5.3] by taking R = J, q = x. The result also gives, in the classical case, the annihilator of a Young permutation module over the integral symmetric group algebra ZSym(n) as a cell ideal (taking R = Z, q = 1).

We return to the the assumption that R is an arbitrary commutative ring and q a unit. For a composition α we write M(α)R,q for the left Hec(n)R,q induced from the trivial module (free over rank 1 over R) for Hec(α)R,q. Then M(α)R,q may be realised as the left ideal Hec(n)R,qx(α) and is isomorphic to M(λ)R,q, where λ is the partition of n obtained by arranging the parts of α in descending order.

Lemma 8.6. Let λ, µ ∈ Par(n) and suppose that µ is a coarsening of λ. Then the annihilator of M(λ)R,q annihilates M(µ)R,q.

Proof. Let A = Hec(n)R,q. For some positive integer m there is a chain λ = ξ(0),ξ(1),...,ξ(m)= µ of elements of Par(n) such that ξ(i) is a simple coarsening of ξ(i − 1), 1 ≤ i ≤ m. If the annihilator of M(ξ(i − 1))R,q is contained in the annihilator of M(ξ(i))R,q, for 1 ≤ i ≤ m then we have

AnnA(M(λ)R,q) = AnnA(M(ξ(0))R,q) ⊆ AnnA(M(ξ(1)R,q)

⊆···⊆ AnnA(M(ξ(m)R,q) = AnnA(M(µ)R,q).

Thus we may assume that µ is a simple coarsening of λ. Writing λ = (λ1, λ2,...), the partition µ has the same parts as the composition α = (λ1, . . . , λi−1, λi + λi+1, λi+2,...), for some i. Now Hec(λ)R,q ⊆ Hec(α)R,q so we have a Hec(λ)R,q-module homomorphism φ : Rx(λ) → M(α)R,q taking x(λ) to x(α) and by Frobenius reciprocity this extends to an A-module homomorphism φ˜ : M(λ)R,q → M(α)R,q. Since x(α) is in the image the map φ˜ is surjective and hence AnnA(M(λ)R,q) ⊆ AnnA(M(α)R,q) and since M(α)R,q is isomorphic to M(µ)R,q we have AnnA(M(λ)R,q) ⊆ AnnA(M(µ)R,q). Recall that for σ ⊆ Par(n) we are writingσ ˆ for the coarsening of σ, i.e. the set of all partitions µ such that µ is a coarsening of some λ ∈ σ.

(dλ) Lemma 8.7. Let σ be a subset of Par(n) and let X = λ∈σ M(λ)R,q , ˆ where dλ 6= 0 for all λ ∈ σ. Then AnnHec(n)R,q (X) = AnnL Hec(n)R,q (X), ˆ where X = λ∈σˆ M(λ)R,q.

Proof. Let AL= Hec(n)R,q. We must show that AnnA(X) annihilates M(µ)R,q for all µ ∈ σˆ. Now such µ is a coarsening of some λ ∈ σ and hence AnnA(X) ⊆ AnnA(M(λ)R,q) ⊆ AnnA(M(µ)R,q), by Lemma 8.6.

36 Recall, from [19], that we have on Hec(n)R,q the R-bilinear symmetric bilinear form h , i : Hec(n)R,q × Hec(n)R,q → R. For a, b ∈ Hec(n)R,q, the value of ha, b i is the coeﬃcient of 1 in ab∗, expressed as an R-linear combination of the elements Tw, w ∈ Sym(n). If s is a standard λ tableau we write [s]= λ. For λ ∈ Par(n), for s a standard λ-tableau and 1 ≤ m ≤ n we write s ↓ m for the the standard tableau whose Young diagram has boxes (i, j) such that the s(i, j) ∈ {1,...,m}, and (s ↓ m)(i, j)= s(i, j). As in [19], we have the dominance partial order on λ tableaux. Thus s E t if [s ↓ m] E [t ↓ m] for all 1 ≤ m ≤ n. For λ tableaux s, t, u, v we write (s,t) E (u, v) if s E u and t E v. Let t be a λ-tableau. We write t′ for the transpose tableau, i.e. the λ′-tableau with t′(i, j) = t(j, i), for (i, j) in the Young diagram of λ′. For standard λ-tableaux s, t, u, v we have ♯ hxs′t′ ,xuvi = 0 unless (u, v) E (s,t) (4) and ♯ N hxs′t′ ,xsti = ±q for some positive integer N (5) by [19, Lemma 4.6]. (dλ) Note that if X = λ∈Par(n) M(λ)R,q then the dλ are uniquely determined by X. (If F is any residue ﬁeld of R and q is the image of q in F then L (dλ) we then we have the Hec(n)F,q-module XF = λ∈Par(n) M(λ)F,q and so the d are uniquely determined by Lemma 5.6). Thus we may extend the λ L notation of Section 5 to Young permutation modules over Hec(n)R.q by putting ζ(X)= {λ ∈ Par(n) | dλ 6= 0}.

Proposition 8.8. Let X be a Young permutation module for Hec(n)R,q and put σ = ζ(X). If σˆ is cosaturated then AnnHec(n)R,q (X) is the cell ideal Hec(n)R,q(τ), where τ is the complement of σˆ. Remark 8.9. One gets the result for “most ” coefficient rings using the main results of this paper. For example if R = F [x,x−1] is the ring of Laurent polynomials over a field F then the result holds for Hec(n)R,x(n) by Lemma 8.4, since Hec(n)F (x),x is semisimple. Moreover for each residue −1 field E of F [x,x ] we have that E ⊗R AnnHec(n)R,x (X) → AnnHec(n)E,q (XE) (dλ) is an isomorphism, by Lemma 5.9. Writing X = λ∈Λ M(λ)R,q and putting ˜ and X = λ∈σ M(λ)F [x,x−1],x we have, for anyL commutative F -algebra R − ˜ and unit q ∈ R, that the natural map R ⊗ 1 Ann − (X) → L F [x,x ] Hec(n)F [x,x 1],x

AnnHec(n)R,q (X) is an isomorphism, by Corollary 2.13. Hence

− ˜ Ann (X)= R ⊗ 1 Ann − (X) Hec(n)R,q F [x,x ] Hec(n)F [x,x 1],x

= R ⊗F [x,x−1] Hec(n)F [x,x−1],x(τ)

= Hec(n)R,q(τ).

37 Likewise if q = 1 then Hec(n)R,q = RSym(n) and the result holds for ZSym(n), since QSym(n) is semisimple. But we have base change on annihilators by Proposition 5.9 and Lemma 3.5. Thus the result holds for RSym(n) for an arbitrary commutative ring R. However, we are unable to get the result for arbitrary R,q by base change from the case J = Z[x,x−1], q = x using the results of Part I since we are assuming there that the base ring is a Dedekind domain (and J is not.) For complete generality we need to use the argument of H¨arterich, [12, Lemma 3] (see also [2, Proposition 7.2]).

Proof. We put A = Hec(n)R,q. By base change and Lemma 8.4 we have A(τ) ⊆ AnnA(X). We must prove AnnA(X) ⊆ A(τ). For a subset ρ of Par(n) we put X(ρ)= λ∈ρ M(λ)R,q. Clearly we may assume X = X(σ). Moreover by Lemma 8.7, we may replace σ byσ ˆ, i.e. we may assume σ is cosaturated. We prove L the result for cosaturated subsets σ of Par(n) by induction on |σ|. The result is clear if σ is empty. Now suppose that σ is not empty and the result holds for smaller cosaturated subsets. Let λ be a minimal element of σ. By the inductive hypothesis the annihilator of X(σ\{λ}) is A(τ {λ}) = A(τ) ⊕ A(λ). Thus we may write h ∈ Ann (X) in the form h = h + h with h ∈ A(τ), h ∈ A(λ) A S 1 2 1 2 and since h1 ∈ AnnA(X) we have h2 ∈ AnnA(X). Thus it suﬃces to prove that h2 = 0, i.e. that an element h ∈ AnnA(X) A(λ) is 0. We ♯ write h = r x ′ ′ , with s,t running over the standard λ tableaux and s,t st s ,t T r ∈ R. Now hM(λ) = 0 i.e. hAx(λ) = 0 and so hT ∗ x(λ)T = 0 st P R,q w1 w2 for all w1, w2 ∈ Sym(n). In particular we have hxuv = 0 for all standard λ ∗ tableaux u, v and hence hxuv = hxvu = 0 for all standard λ tableaux u, v. Thus we have hh, xuvi = 0 i.e.

♯ rsthxs′t′ ,xuvi = 0 s,t X for all standard λ tableaux u, v. Suppose, for a contradiction that h 6= 0 and choose a pair of λ tableaux (s0,t0) maximal (in the ordering E) in the set {(s,t) ∈ N˜(λ) | rst 6= 0}. ♯ ♯ ′ ′ ′ ′ Then s,t rsthxs t ,xs0t0 i = 0. But if a summand rsthxs t ,xs0t0 i is non-zero then we have r 6= 0 and (s ,t ) E (s,t), by (4), and hence (s,t) = (s ,t ). P st 0 0 0 0 Hence we have

♯ ♯ ′ ′ ′ ′ rsthx ,xs0t0 i = rs0t0 hx ,xs0t0 i s t s0t0 s,t X N which, by (5), is ±q rs0t0 , for some positive integer N, and so rs0t0 = 0, a contradiction. Hence h = 0 and AnnA(X)= A(τ).

38 Example 8.10. Let H be a hook module for Hec(n)R,q, i.e. a ﬁnite di- (dλ) rect sum λ M(λ)R,q , with λ running over hook partitions of n. Recall that the index index(H) of H is the smallest positive integer a such that L d(a,1n−a) 6= 0. Then putting σ = {λ ∈ Par(n) | dλ 6= 0} the set σˆ consists of all λ = (λ1, λ2,...) ∈ Par(n) such that λ1 ≥ index(H), see Section 6.

Hence AnnHec(n)R,q (H) is the cell ideal Hec(n)R,q(τ), where τ is the set of partitions λ = (λ1, λ2,...) such that λ1 < index(H). In particular we can take q = 1. Let V be a free module of rank n over an arbitrary commutative ring R, with R-basis v1, . . . , vn. We regard V , and ⊗r hence the r-fold tensor product V = V ⊗R ⊗···⊗R V as a permutation module for A = RSym(n). Then V ⊗r is the permutation module on the Sym(n)-set I(n, r). We regard V ⊗r as an RSym(m)-module, for m ≤ n, by ⊗r restriction. Then from the above (or Remark 8.9) AnnRSym(m)(V ) is the cell ideal A(τ), where τ is the set of all λ = (λ1, λ2,...) ∈ Par(m) such that λ1 < index(I(n, r)|Sym(m)), i.e. (by Section 6, (2)) λ1 < m − min{m, r}. That this holds for m = n,n − 1 and with R a commutative domain is the main result Theorem 7.5 of [2].

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