On Schur algebras and related algebras VI: Some remarks on rational and classical Schur algebras

Journal of Algebra 405 (2014) 92–121

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On Schur algebras and related algebras VI: Some remarks on rational and classical Schur algebras Stephen Donkin a r t i c l e

i n f o

Article history: Received 24 May 2012 Available online 4 March 2014 Communicated by Volodymyr Mazorchuk Keywords: Algebraic groups Schur algebras Representation theory Classical groups

a b s t r a c t In a recent paper, Dipper and Doty, [4], introduced certain finite dimensional algebras, associated with the natural module of the general linear group and its dual, which they call rational Schur algebras. We give a proof, via tilting modules, that these algebras are in fact generalised Schur algebras. Using the same technique we show that certain finite dimensional algebras associated with classical groups, introduced by Doty, [20], are quasi-hereditary algebras. A generalised Schur algebras may be viewed as a quotient of the algebra of distributions of a reductive group by a certain ideal. We give generators for this ideal. © 2014 Elsevier Inc. All rights reserved.

Introduction Let k be an infinite field, let G(n) = GLn (k), let E be the natural kG(n)-module (of column vectors) and denote by E ∗ the dual module. Let r, s be non-negative integers. Then the coefficient space cf(E ⊗r ⊗ E ∗⊗s ) is a certain finite dimensional subcoalgebra of the coordinate algebra k[G(n)]. In a recent paper, [4], Dipper and Doty study the dual algebra, which they denote S(n; r, s) and call the rational Schur algebra (one obtains the ordinary Schur algebra, as in Green’s monograph, [24], by taking s = 0). One of the main results of their paper is that this algebra is a generalised Schur algebra, in the sense of [9–11,13,14,16–19]. Since every generalised Schur algebra is quasi-hereditary, [9, (2.2)], in particular S(n; r, s) is. However, there is a gap in the argument given in [4]. One must show that S(n; r, s) is the generalised Schur algebra S(π) for the appropriate http://dx.doi.org/10.1016/j.jalgebra.2014.01.034 0021-8693/© 2014 Elsevier Inc. All rights reserved.

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finite saturated set of dominant weights π. The Schur algebra S(π) is realised as a quotient algebra Uk /I (cf. [9, 3.2] and [13]). Here I is the ideal of the hyperalgebra Uk (or algebra of distributions) consisting of the elements that vanish on all finite dimensional Uk -modules having all composition factors in {L(λ) | λ ∈ π}, where L(λ) is the simple module with highest weight λ. In [4], the algebra S(n; r, s) is realised as a certain quotient Uk /I  for an ideal I  of Uk containing I. It is shown, [4, p. 71], that I  annihilates all irreducible modules L(λ), with λ ∈ π and it is claimed that this implies I   I. However, that I  annihilates all the irreducible modules of Uk /I shows only that I  /I is contained in the Jacobson radical of Uk /I and so this argument is valid only when S(π) = Uk /I is semisimple. In [4, Section 5], it is pointed out that the fact that S(n; r, s) is quasihereditary can also be deduced from the cellular structure described in [4, Section 6]. However, the argument for cellularity given there, [4, p. 75], relies on the independence of the base field of the dimension of S(n; r, s) and this is justified via the realisation of S(n; r, s) as a generalised Schur algebra. Working over the ground field Q, an algebra is described in [4, Section 7.3], by generators and relations and it is claimed that this gives a presentation of S(n; r, s). However, the algebra described there is not isomorphic to S(n; r, s) and in fact gives a different generalised Schur algebra (see the Remark of Section 3 below). In Section 1, we prove that the algebras S(n; r, s) are generalised Schur algebras via a different approach using a method described in [18] that relies on properties of tilting modules. Since the argument is no more difficult for the quantum situation we work with quantum general linear groups over an arbitrary non-zero parameter q in an arbitrary base field in this section. Let π be a finite (not necessarily saturated) set of dominant weights. Let I(π) be the ideal of Uk consisting of the elements of Uk that annihilate every finite dimensional Uk -module having all composition factors in {L(λ) | λ ∈ π}. We show that I(π) is generated by I(π) ∩ Uk (h), where Uk (h) is the hyperalgebra (or algebra of distributions) of a maximal torus. The case in which π is saturated and k has characteristic 0 is shown by Doty, Giaquinto and Sullivan in [22, 5.3 Theorem]. Applying this to the case in which G is a general linear group and k has characteristic 0, we deduce a result equivalent to the description of Doty and Giaquinto, [21, Theorem 2.1] of the Schur algebra S(n, r) by generators and relations. We give also the appropriate generalisation for S(n; r, s). In order to make this paper self contained we include relevant definitions and elementary arguments even when there is some overlap with [4]. (Compare for example Section 1.1 below and [4, Section 4].) Finally, we note that a double centraliser theorem for the rational Schur algebra over an arbitrary infinite field has been obtained by Tange, [27, Theorem 4.1]. For an earlier treatment of this in a more general context see the papers by Dipper, Doty and Stoll, [5,6]. For related recent work see also [2].

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1. S(n; r, s) as a generalised Schur algebra 1.1. We start with some combinatorics that underly the representation theory of general linear groups. Let n be a positive integer. We set X(n) = Zn and call the elements of X(n) weights. A weight λ = (λ1 , . . . , λn ) is called dominant if λ1  λ2  · · ·  λn . We write X + (n) for the set of all dominant weights. We set i = (0, . . . , 0, 1, 0, . . . , 0) ∈ X + (n) (where i has 1 in the ith position). We write Π for the set of simple roots 1 −2 , 2 −3 , . . . , n−1 −n . We write Φ+ for the set of positive roots i −j , 1  i < j  n, and write Φ for the set of all roots i −j , 1  i, j  n, i = j. A weight λ = (λ1 , . . . , λn ) is called polynomial if λ1 , . . . , λn  0. We write Λ(n) for the set of all polynomial weights. We write Λ+ (n) for X + (n) ∩ Λ(n) and identify Λ+ (n) with the set of partitions with at most n parts. For λ ∈ Zn we define |λ| = |λ1 | + · · · + |λn |. If all λi  0 then we call |λ| the degree of λ. For a non-negative integer r we write Λ+ (n, r) for the set of λ ∈ Λ+ (n) such that |λ| = r. We identify Λ+ (n, r) with the set of partitions of r into at most n parts. We partially order X(n) by decreeing that λ  μ if μ − λ is a sum of positive roots. More concretely, we have λ = (λ1 , . . . , λn )  μ = (μ1 , . . . , μn ) if and only if λ1 +· · ·+λi  μ1 + · · · + μi , for 1  i < n and λ1 + · · · + λn = μ1 + · · · + μn . Note that r1 is the unique maximal element of Λ+ (n, r). A subset π of X + (n) will be called saturated if whenever λ ∈ π and μ ∈ X + (n) is such that μ  λ then μ ∈ π. The smallest saturated subset of X + (n) containing an arbitrary subset π of X + (n) will be called the saturation of π. For λ = (λ1 , . . . , λn ) ∈ X(n) we define the dual weight λ∗ to be (−λn , . . . , −λ1 ). Note that λ is dominant if and only if λ∗ is dominant and that λ  0 if and only if λ∗  0. + + For λ = (λ1 , . . . , λn ) ∈ X + (n) we define λ+ = (λ+ 1 , . . . , λn ) ∈ X (n) by  λ+ i

=

λi ,

if λi  0;

0,

if λi < 0,

− + and define λ− = (λ− 1 , . . . , λn ) ∈ X (n) by

λ− i =



λi ,

if λi  0;

0,

if λi > 0.

Thus we have λ = λ+ + λ− and λ+ , λ− ∈ X + (n). Moreover, λ+ and (λ− )∗ are partitions. We fix non-negative integers r, s. We define Λ+ (n; r, s) to be the set of all λ ∈ X + (n) such that |λ+ |  r, |λ− |  s and r − |λ+ | = s − |λ− |. (1) Assume n  2. Then Λ+ (n; r, s) is the saturation of r1 − sn . Proof. Set π = Λ+ (n; r, s). Suppose that λ = (λ1 , . . . , λn ) ∈ π and a = r−|λ+ | = s−|λ− |. Then (r − a)1 − (s − a)n  r1 − sn since

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  (r1 − sn ) − (r − a)1 − (s − a)n = a(1 − n ) and 1 − n ∈ Φ+ . Since λ+ is a partition of r − a we have λ+  (r − a)1 . Also, we have (−(s − a)n − λ− )∗ = (s − a)1 − (λ− )∗ and since (λ− )∗ is a partition of s − a, we have (s−a)1 −(λ− )∗  0 and hence (−(s−a)n −λ− )∗  0 and therefore −(s−a)n −λ−  0. Thus we have (r − a)1 − (s − a)n  λ+ + λ− = λ and hence r1 − sn  (r − a)1 − (s − a)n  λ. Hence λ ∈ Λ+ (n; r, s) belongs to the saturation of r1 − sn . We now show that π is saturated. Assume for a contradiction that it is not. Let μ ∈ X + (n) be maximal such that μ  λ for some λ ∈ π but μ ∈ / π. Then |λ+ | = r − a − and |λ | = s − a for some a. By maximality we must have μ = λ − i + j , for some i < j, see [26, I, (1.16)]. Let 1  m  n be such that λm  0 and λt < 0 for t > m. If j  m then we have μ+ = λ+ − i + j , μ− = λ− and |μ+ | = r − a, |μ− | = s − a. Similarly, if i > m then μ+ = λ+ and μ− = λ− − i + j and |μ+ | = r − a, |μ− | = s − a. If i  m and j > m then we have μ+ = λ+ − i , μ− = λ− + j and |μ+ | = r − a − 1, |μ− | = s − a − 1. Hence μ ∈ π. 2 1.2. We set up the representation theory of reductive groups (see [18] for fuller details) and state a criterion (the Proposition below) that will be useful to us in several contexts. We begin by recalling some definitions and constructions in the coalgebra context. For further details see [23] and [9, Section 1.2]. So let k be a field and let (A, δ, ) be a k-coalgebra. By an A-comodule we mean a right A-comodule. For an A-comodule (V, τ ) with basis vi , i ∈ I, we have the “coefficient functions” fij defined by the equations τ (vi ) =

n 

vj ⊗ fji .

j∈I

The subspace cf(V ) of k[G] spanned by the elements fij is independent of the choice of basis and called the coefficient space of V . Let {L(λ) | λ ∈ Λ} be a complete set of pairwise non-isomorphic simple A-comodules. Let π be a subset of Λ. We say that an A-comodule V belongs to π if for each composition factor L of V we have that L is isomorphic to L(λ) for some λ ∈ π. For any A-comodule V there is, among all subcomodules belonging to π, a unique maximal one, which we denote Oπ (V ). Regarding A itself as a right A-comodule (with structure map δ : A → A ⊗ A) we have the subcomodule A(π) = Oπ (A). In fact A(π) is a subcoalgebra of A and we have A(π) =  V cf(V ), with the sum running over all finite dimensional A-comodules belonging to π,

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see the argument of [9, Section 1.2], or explicitly [25, II, A.14, (2)] (in the context of algebraic groups and their representations, but the argument is general). We now suppose that k is algebraically closed. If H is a linear algebraic group over k with coordinate algebra k[H] then k[H] is naturally a Hopf algebra over k and the category of rational left H-modules is equivalent to the category of right k[H]-comodules so that the above discussion applies to rational representations of H. Now let G be a reductive algebraic group over k. Let T be a maximal torus of G and W = NG (T )/T be the corresponding Weyl group (where NG (T ) denotes the normaliser of T in G). Then W acts naturally on the character group X(T ) of T and hence on R ⊗Z X(T ). Let ( , ) be a W -invariant, positive definite symmetric bilinear form on R ⊗Z X(T ). The Lie algebra Lie(G) is naturally a rational T -module. Let Φ denote the set of non-zero weights. Then (V, Φ) is a root system, where V denotes the R-span of Φ in R ⊗Z X(T ). Let B be a Borel subgroup of G. The set Φ− of non-zero weights for the action of T on Lie(B) is a system of negative roots. Let Φ+ = {−α | α ∈ Φ− } (a system of positive roots) and let Π be the set of simple roots. Let X + (T ) = {λ ∈ X(T ) | (λ, α)  0 for all α ∈ Φ+ }, the set of dominant weights. For λ ∈ X(T ) we have the one dimensional B-module kλ on which T acts with weight λ. For λ ∈ X + (T ) the induced module ∇(λ) = indG B kλ has simple socle L(λ) + and {L(λ) | λ ∈ X (T )} is a complete set of pairwise non-isomorphic simple rational G-modules. The module L(λ) has unique highest weight λ (for λ ∈ X + (T )). We shall also need the Weyl modules. For λ ∈ X + (T ) we define the Weyl modules Δ(λ) to be the G-module dual of ∇(λ∗ ), where λ∗ = −w0 λ, and w0 is the longest element of W . For λ ∈ X + (T ) the modules ∇(λ) and Δ(λ) have the same composition factors, counted according to multiplicity. For λ ∈ X + (T ) both ∇(λ) and Δ(λ) have character χ(λ) given by Weyl’s character formula, see [25, II, 5.10]. Let π be a subset of X + (T ). Taking A = k[G] we have the subcoalgebra A(π) of k[G]. If π is finite then A(π) is finite dimensional and the dual algebra S(π) has the property that the category of rational left G-modules which belong to π is naturally equivalent to the category of all left S(π)-modules. By a good filtration of a finite dimensional rational G-module M we mean a filtration 0 = M0  M1  · · ·  Mr = M such that, for each 0 < i  r, the quotient Mi /Mi−1 is either 0 or is isomorphic to ∇(μ), for some μ ∈ X + (T ) (which may depend on i). A finite dimensional rational G-module M is called a tilting module if M admits a good filtration and also the dual module M ∗ admits a good filtration. For each λ ∈ X + (T ), there is an indecomposable tilting module M (λ), say, which has highest weight λ, and moreover {M (λ) | λ ∈ X + (T )} is a complete set of pairwise non-isomorphic indecomposable tilting modules. We shall assume the notation of our survey article on tilting modules for algebraic groups [18], except that we write M (λ) for the tilting module for G which was denoted there by T (λ). The following was noted in [18, 7.3 Corollary], as a consequence of the more general result [18, Theorem 7.1] (taken from [12, Theorem of Section 1]).

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Proposition. For a saturated subset π of X + the generalised Schur coalgebra A(π) is spanned by the subspaces cf(M (λ)), λ ∈ π. Remark. The natural context for the above Proposition, at least for π finite, is really the theory of quasi-hereditary algebras, as set up by Cline, Parshall and Scott, [3]. We make some remarks on the transition to this situation. We refer the reader to the Appendix of [17] for an exposition of the theory of quasi-hereditary algebras suitable for our purposes here and in particular the equivalence of this theory and the theory of highest weight categories. Let k be a field and S a finite dimensional algebra. We write mod(S) for the category of finite dimensional left S-modules and for V ∈ mod(S) write AnnS (V ) for the annihilator or V in S. Now suppose that S is a (finite dimensional) quasi-hereditary algebra over a field k. We assume that S is Schurian, in the sense that EndS (V ) = k for every simple module V . Let {L(λ) | λ ∈ Λ} be a complete set of pairwise non-isomorphic simple modules, where Λ is a partially ordered set with respect to which the category mod(S) of finite dimensional left S-modules is a highest weight category. For λ ∈ Λ let Δ(λ) be the corresponding standard module, let ∇(λ) be the corresponding costandard module and let M (λ) be the corresponding indecomposable tilting module. Let π be a subset of Λ. We say that a left S-module belongs to π if all of its composition factors belong to {L(λ) | λ ∈ π}. Let I(π) be the ideal consisting of the elements of S than annihilate every left S-module belonging to π and let S(π) = S/I(π). We regard an S-module annihilated by I(π) as an S(π)-module in the natural way. The modules {L(λ) | λ ∈ π} form a complete set of pairwise non-isomorphic irreducible modules for π. Assume now that π is saturated. Then S(π) is quasi-hereditary, for the partial order on π induced from that on Λ. Furthermore, for λ ∈ π, we have that Δ(λ) is the corresponding standard module, ∇(λ) is the corresponding costandard module and M (λ) is the corresponding indecomposable tilting module. Let V ∈ mod(S(π)). Then V has ∇-filtration if an only if V has finite left resolution 0 → Tm → Tm−1 → · · · → T0 → V → 0 by tilting modules, see e.g. [17, Proposition A4.4]. (The argument first appeared in the context of rational representation of linear algebraic groups, see [8, Section 1].) By the dual argument one also has that V has a Δ-filtration if and only if V has a finite right resolution by tilting modules 0 → V → T0 → T1 → · · · → Tn → 0. In particular if V ∈ mod(S(π)) has a Δ-filtration then there exists an embedding of V into a tilting module for S(π). We apply this to the left regular module S(π) S(π), which has a Δ-filtration since S(π) is quasi-hereditary. Thus we have an embedding S(π) S(π) → M where M is a direct sum of copies of M (λ), λ ∈ π.

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Hence the annihilator of M in S is contained in the annihilator of S(π) = S/I(π) in S, i.e. I(π). The module M is a direct sum of indecomposable tilting modules M (λ),  λ ∈ π, so we get λ∈π AnnS (M (λ)) ⊆ I(π). The reverse inclusion holds by the definition of I(π) so we have: (1) I(π) =

 λ∈π

AnnS (M (λ)).

Suppose the algebra S arises as the dual A∗ of a coalgebra A. We identify right A-comodules and left S-modules in the usual way. For a subspace A0 of A and a subspace S0 of S we define the perpendicular spaces 

A⊥ 0 = s ∈ S s(a) = 0 for all a ∈ A0 and 

S0⊥ = a ∈ A s(a) = 0 for all s ∈ S0 . Let V be a finite dimensional S-module (equivalently A-comodule) with coefficient space cf(V ). (See Green, [23] for a discussion of coefficient spaces in the context of the representation theory of coalgebras.) Then it is easy to check that cf(V )⊥ = AnnS (V ) and so cf(V ) = AnnS (V )⊥ . Taking perpendicular spaces in (1) we get ⊥

I(π) =

 λ∈π



AnnS M (λ)

 ⊥

=



AnnS M (λ)

⊥

.

λ∈π

Moreover, we have I(π)⊥ = A(π) and AnnS (M (λ))⊥ = cf(M (λ)), for λ ∈ Λ. Thus we have: (2) A(π) =

 λ∈π

cf(M (λ)).

The transition from a finite saturated set π to an arbitrary saturated set, as in the Proposition, is done by a local finiteness argument, as indicated in the proof of the Theorem of Section 1 of [12]. 1.3. We now focus on the quantum general linear case. We allow k to be an arbitrary field. By abuse of notation (and by analogy with the theory of algebraic groups) we indicate by the phrase “G is a quantum group over k”, that we have in mind a Hopf algebra over k, which we denote k[G] and call the coordinate algebra of G. We indicate by the phrase “H is a (quantum) subgroup of G” that we have in mind a Hopf ideal of k[G] denoted IH , called the defining ideal of H and that H is the quantum group with coordinate algebra k[H] = k[G]/IH .

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Let G be a quantum group over k. By a left G-module we mean a right k[G]-comodule and by a morphism of left G-modules we mean a morphism of right k[G]-comodules. We write mod(G) for the category of finite dimensional left G-modules. Let 0 = q ∈ k. We adopt the notation of [17, 0.20], and write A(n) for the (usually non-commutative) k-bialgebra described there by generators and relations. Thus A(n), as a k-algebra is generated by elements cij , 1  i, j  n. We have the determinant element d of A(n) defined, as usual, by the equation d=



sgn(ρ)c1,1ρ c2,2ρ . . . cn,nρ

ρ∈Sym(n)

(though the order of the terms in each summand is important) where Sym(n) denotes the symmetric group on {1, 2, . . . , n} and sgn(ρ) denotes the sign of a permutation ρ. The Ore localisation A(n)d of A(n) at d has a natural Hopf algebra structure. We write G(n) for the quantum general linear group of degree n, i.e., the quantum group with coordinate algebra k[G(n)] = A(n)d . The standard maximal torus T (n) in G(n) is by definition the quantum subgroup with defining ideal I(n) generated by all cij with i = j. Then k[T (n)] is the (commutative) −1 algebra of Laurent polynomials k[x1 , x−1 1 , . . . , xn , xn ], where xi = cii +IT (n) and comultiplication k[T (n)] → k[T (n)]⊗k[T (n)] takes xi to xi ⊗xi . For λ = (λ, . . . , λn ) ∈ X(n) we set xλ = xλ1 1 . . . xλnn . Let V be a T (n)-module with structure map τ : V → V ⊗ k[T (n)]. For λ ∈ X(n) we have the weight space 

V λ = v ∈ V τ (v) = v ⊗ xλ and we have the weight space decomposition V =



V μ.

μ∈X(n)

We write E for the natural G(n)-module. Thus E has k-basis e1 , . . . , en and comodule n structure map τ : E → E ⊗ k[G(n)] taking ei to j=1 ej ⊗ cji , for 1  i  n. Let E ∗ denote the dual G(n)-module. Recall that for each λ ∈ X + (n) we have the induced module ∇(λ) (see e.g., [16]). As in the cases considered in Section 1.2, an ascending filtration 0 = V0  V1  · · ·  Vm of V ∈ mod(G) is called good if for each 0 < i  m we either have Vi = Vi−1 or Vi /Vi−1 is isomorphic to ∇(μ), for some μ ∈ X + (T ) (which may depend on i). A finite dimensional G(n)-module V is called a (partial) tilting module if both V and the dual module V ∗ admit good filtrations. For each λ ∈ X + (n) there is an indecomposable tilting module M (λ) ∈ mod(G) that has highest weight λ. Moreover, λ is the unique highest weight of M (λ) and occurs with multiplicity 1. Furthermore {M (λ) | λ ∈ X + (n)} is a complete set of pairwise non-isomorphic tilting modules for G(n) (see e.g., [16, Section 4, (3)(ii)]). The class of tilting modules is closed under the

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operations of taking direct sums, direct summands and tensor products. Moreover, for r r each r  0, the exterior power E is a tilting module, as is its dual ( E)∗ . We consider the category mod(G(n), π) of modules belonging to π, i.e., the full subcategory of mod(G(n)) whose objects are the G-modules V such that every composition factor of V belongs to {L(λ) | λ ∈ π}. For a finite sequence of non-negative integers λ = (λ1 , . . . , λm ) we define λ

E=

λ1

E ⊗ ··· ⊗

 λm

E.

We denote by λ the transpose of a partition λ. If λ = (λ1 , . . . , λm ) is a partition and λ E has unique highest weight λ and this weight occurs with multiplicity λ1  n then one. Similarly for a finite sequence of non-negative integers μ = (μ1 , . . . , μm ) we define μ ∗ μ μm ∗ μ ∗ E = ( 1E ⊗ · · · ⊗ E) . If μ is a partition and μ1  n then E has unique highest weight (μ )∗ and this weight occurs with multiplicity one. Hence we have the following result. (1) Let λ ∈ Λ+ (n; r, s). Define partitions μ and τ by μ = λ+ and τ  = (λ− )∗ . Then μ τ E ⊗ E ∗ has unique highest weight λ and the λ weight space is one dimensional. Hence we have μ

E⊗

τ

E ∗ = M (λ) ⊕ N

where N is a direct sum of tilting modules M (ξ), with ξ < λ. This quickly leads to the following result. (2) For λ ∈ Λ+ (n; r, s) we have cf(M (λ))  cf(E ⊗r ⊗ E ∗⊗s ). Proof. Let μ, τ be as in (2). Then we have μ τ    cf M (λ)  cf E⊗ E∗ so it suffices to prove that cf

μ

E⊗

τ

   E ∗  cf E ⊗r ⊗ E ∗⊗s .

We have |μ| = r−a, |τ | = s−a for some a  0. Moreover, we have a natural G-module surμ μ jection E ⊗(r−a) → E from the definition and hence cf( E)  cf(E ⊗(r−a) ). More τ τ ⊗(s−a) over, the natural surjection E → E induces an injection ( E)∗ → (E ⊗(s−a) )∗ . Hence we have cf

 τ

  ∗  E ∗  cf E ⊗(s−a) .

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Thus we have cf(M (λ))  cf(E ⊗r−a ).cf((E ⊗s−a )∗ ) and it suffices to prove that   ∗     ∗    cf E ⊗(r−a) ⊗ E ⊗(s−a) = cf E ⊗(r−a) .cf E ⊗(s−a)  cf E ⊗r ⊗ E ∗⊗s . ∼ E ∗⊗(s−a) (since G(n)-modules with the same weights Moreover, we have (E ⊗(s−a) )∗ = spaces, counting multiplicities, are isomorphic, by the classification of tilting modules by highest weight). So it suffices to prove that       cf E ⊗(r−a) .cf E ∗⊗(s−a)  cf E ⊗r ⊗ E ∗⊗s . Thus it suffices to prove that cf(E ⊗i ).cf(E ∗⊗j ) is contained in cf(E ⊗(i+1) ⊗ E ∗⊗(j+1) ), for i, j  0. However, we have the natural injection k → E ⊗ E ∗ so that E ⊗i ⊗ E ∗⊗j ∼ = ⊗i ∗⊗j embeds in E ⊗k⊗E E ⊗r ⊗ E ⊗ E ∗ ⊗ E ∗⊗s = E ⊗(i+1) ⊗ E ∗⊗(j+1) so this clear. 2 From 1.2 Proposition and the Remark following it we now get A(π)  cf(E ⊗r ⊗E ∗⊗s ). But also E ⊗r ⊗ E ∗⊗s is a module belonging to π so that cf(E ⊗r ⊗ E ∗⊗s )  A(π) (see the first paragraph of Section 1.2). (3) For π = Λ+ (n; r, s) we have A(π) = cf(E ⊗r ⊗ E ∗⊗s ). The algebra S(n; r, s) is by definition the dual algebra of the coalgebra cf(E ⊗r ⊗ E ∗⊗s ) and the generalised Schur algebra S(n; r, s) is by definition the dual algebra of the coalgebra A(π) so we have: (4) For π = Λ+ (n; r, s) we have S(π) = S(n; r, s), in particular S(n; r, s) is a generalised Schur algebra. 2. The classical groups 2.1. In [20], Doty introduces a finite dimensional algebras Sr (G), defined by a classical group G and a non-negative integer r. In type A one gets the usual Schur algebras, as in Green’s monograph, [24]. We show that most of these are generalised Schur algebras in the sense [9,10] (as expected by Doty, [20, p. 167]). The type A case was the motivation for the general definition (of generalised Schur algebra) introduced in [9]. We shall argue here via 1.2 Proposition. However, we start with a generality that will be useful in our discussion of type B. Let S be a quasi-hereditary algebra with partially ordered set Λ and labelling of simple modules L(λ), λ ∈ Λ, which makes mod(S) into a highest weight category. Let π be a subset of Λ. An S module belongs to π if all of its composition factors belong to {L(λ) | λ ∈ π}. We write mod(π) for the full subcategory of mod(S) whose objects

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are the modules belonging to π. Let I(π) be the ideal consisting of the elements of S that vanish on all S-modules belonging to π and let S(π) = S/I(π). We are interested in conditions that ensure that mod(π) is a highest weight category (equivalently S(π) is a quasi-hereditary algebra). Of course this is the case if π is saturated (see e.g. [17, Appendix, Proposition A3.4]). However, this condition is much too restrictive in practice. For example, a block of a quasi-hereditary algebra is naturally a quasi-hereditary algebra (and the set of λ ∈ Λ such that L(λ) belongs to a given block is not in general saturated). We give the following criterion. Proposition. Suppose that π is a subset of Λ with the condition that whenever λ ∈ π and μ ∈ Λ is such that L(μ) is a composition factor of ∇(λ) or Δ(λ) then μ belongs to π. Then mod(π) is a highest weight category and S(π) is a quasi-hereditary algebra with respect to the labelling of simples L(λ), λ ∈ π, and the induced partial order on π. Proof. We show that mod(π) is a highest weight category. Let λ ∈ π. Let I be an S-module of largest possible dimension subject to the requirements that it has simple socle L(λ) and admits a filtration with sections of the form ∇(τ ) with τ ∈ π. Note that I has dimension at least that of ∇(λ) since ∇(λ) satisfies the required conditions. In particular, I is not zero. By the assumptions, I belongs to π. We claim that in fact I is injective in the category mod(π), which clearly implies the result. Assume for a contradiction that this is not the case. Then there is a simple module L belonging to π with Ext1S (L, I) = 0. We choose μ ∈ π minimal such that Ext1S (L(μ), I) = 0. We have a short exact sequence of S-modules 0 → L(μ) → ∇(μ) → Q → 0. Hence we get an exact sequence     Ext1S (Q, I) → Ext1S ∇(μ), I → Ext1S L(μ), I → Ext2S (Q, I).

(∗)

Note that we have Ext1S (Q, I) = 0 since a composition factor of Q has the form L(ν) for some ν ∈ π with ν < μ. We claim also that Ext2S (Q, I) = 0. If not there is a composition factor L of Q with Ext2S (L, I) = 0. Moreover, L has the form L(ν) for some ν < μ. There is a short exact sequence of S-modules 0 → R → Δ(ν) → L(ν) → 0. We have ExtiS (Δ(ν), I) = 0 for i = 1, 2, since I has a ∇-filtration, see e.g. [17, Proposition A.2.2(ii)]. Thus we get an isomorphism Ext1S (R, I) → Ext2S (L(ν), I). But now a composition factor of R has the form L(ξ) with ξ ∈ π and ξ < ν and hence ν < μ. By the minimality assumption we therefore have Ext1S (R, I) = 0 and hence Ext2S (L(ν), I) = 0. Hence Ext2S (Q, I) = 0. Now from (∗) the natural map Ext1S (∇(μ), I) → Ext1S (L(μ), I) is an isomorphism. Hence there is an inclusion of I in an S-module J giving a non-split extension 0 → I → J → ∇(μ) → 0. Let J0 be the submodule of J containing I such that J0 /I is isomorphic to L(μ). Since the map Ext1S (∇(μ), I) → Ext1S (L(μ), I) is an isomorphism the extension 0 → I → J0 → L(μ) → 0 is non-split. Thus the socle of J0 is L(λ) and (by left exactness of the operation of taking the socle) J has simple socle L(λ). But then J contradicts the maximality of I. 2

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In the context of Section 1.2, bearing in mind that the modules ∇(λ) and Δ(λ) have the same composition factors, this specialises to a statement about representations of reductive groups as follows. Corollary. Let π denote a finite set of dominant weights and let A(π) be the sum of all coefficient spaces of rational G-modules belonging to π. Suppose that for λ ∈ π and μ ∈ X + we have [∇(λ) : L(μ)] = 0 for μ ∈ / π. Then the dual algebra S(π) of A(π) is quasi-hereditary with respect to the labelling of simples {L(λ) | λ ∈ π} and the partial order on π induced from X + . To derive the Corollary from the Proposition choose Λ to be a finite saturated subset of X + containing π and take S = S(Λ) and identify S(π) with the quotient S/I(π) as above. 2.2. Now let G be a classical group over an algebraically closed field k, with natural module E. The r-fold tensor product E ⊗r is a G-module hence kG-module. We write Ar (G) for the coalgebra cf(E ⊗r ) and Sr (G) for the dual algebra. The natural action of Sr (G) is faithful and the image of the action is also the image of the representation kG → Endk (E ⊗r ). In this way we identify Sr (G) with the image of the representation kG → Endk (E ⊗r ). It is easy to check that we have a natural isomorphism Sr (G) → Ar (G)∗ . Each of the classical groups has, in our context, its own individual features and in the interests of clarity we treat each case separately. We shall discuss the classical groups in turn in order of difficulty: A, C, D, B. We first make a remark which will be useful in treating types B, C and D. Remark. Let G be a semisimple, connected, classical group of type B, C or D over an algebraically closed field k of characteristic different from 2. The natural module E is self dual so that E ⊗ E contains a copy L, say, of the trivial module. Hence we have k = cf(L)  cf(E ⊗ E). Thus if r and s are non-negative integers and s − r = 2t, say, is an even non-negative integer. Then we have         cf E ⊗r = cf E ⊗r ⊗ L⊗t  cf E ⊗r ⊗ E ⊗2t = cf E ⊗s . 2.3. We first consider type A and take G = GLn (k) acting on its natural representation. The fact that Sr (G) is naturally identified with the Schur algebra S(n, r) is proved in Green’s monograph, [24, Section 2.4]. The fact that Sr (G) = S(n, r) is a generalised Schur algebra was shown in [9]. A different argument, using 1.2 Proposition, was given in [18, p. 243]. 2.4. We now consider type C. For this case a rather complicated argument was given in [11]. The fact that 1.2 Proposition may be used to establish the result in type C was

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indicated in [18, Remark 8.4]. We here add some details to this Remark. We take E to be a finite dimensional k-vector space of even dimension endowed with a (non-singular) symplectic form and write GSp for the subgroup of GL(E) consisting of all symplectic similitudes, as in [20]. We write G for the symplectic group Sp(E). Now GSp is the product of G and the group of scalar matrices. Moreover, each scalar matrix acts on E ⊗r by scalar multiplication so we have Sr (GSp) = Sr (G). We consider the coefficient space of cf(E ⊗r )  k[G] of E ⊗r as a G-module. Let n = 2m be the dimension of E. Let e1 , . . . , en be a basis such that (ei , ej ) is 1 if i < j and i + j = n + 1, is −1 if i > j and i + j = n + 1, and is 0 in all other cases. We identify G with a subgroup of GLn (k), via this choice of basis. Let T be the maximal torus of G consisting of diagonal matrices. For 1  i  n we have the multiplicative character i of T taking an element of T to its (i, i) entry. Then the character group X(T ) is Z-free on 1 , . . . , m . We identify an m-tuple of integers α = (α1 , . . . , αm ) ∈ Zm with the element α1 1 + · · · + αm m of X(T ). Let B be the Borel subgroup of G consisting of lower diagonal matrices. Then the set − Φ of non-zero weights of T acting on the Lie algebra of B forms a system of negative roots in the set of roots Φ of G (with respect to T ). Let ωi = 1 + · · · + i , 1  i  m. Then ω1 , . . . , ωm are the fundamental weights. It was shown in [15, Appendix A], that the restriction of a GL(E)-module with a good filtration has a good filtration as a G-module. Hence the restriction of a tilting module for GL(E) is a tilting module for G. If follows that for a finite sequence of non-negative integers α = (α1 , α2 , . . . , αs ), the G-module α

E=

α1

E⊗

α2

E ⊗ ··· ⊗

αs

E

is a tilting module. We fix r and consider rω1 . Let π = {λ ∈ X + (T ) | λ  rω1 }. Then π consists of all partitions λ with at most m parts, such that |λ| = r − 2s for some integer 0  s  r/2.  λ E, where λ denotes the dual partition of λ. Note that λ ∈ π is the highest weight of λ λ  E, in particular we have cf(M (λ))  cf( E). Hence M (λ) is a direct summand of  λ E, where h = |λ | = |λ| = r − 2s, Moreover, we have the natural surjection E ⊗h → for some 0  s  r/2. Hence we have cf

λ      E  cf E ⊗h  cf E ⊗r

λ (using the Remark of Section 2.2). Thus we get cf(M (λ))  cf( E)  cf(E ⊗r ). Hence ⊗r ⊗r is a G-module belonging by 1.2 Proposition, we have A(π)  cf(E ). Moreover, E to π so we have cf(E ⊗r )  A(π). Hence we have A(π) = cf(E ⊗r ) and therefore S(π) = Sr (G). Thus Sr (G) is the generalised Schur algebra determined by π.

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2.5. We now consider type D. Let E be a k-vector space of even dimension n = 2m  4. We assume the characteristic of k is not 2. We now choose a basis e1 , . . . , en of E and endow E with the bilinear form given by (ei , ej ) = 1 if i + j = n + 1 and (ei , ej ) = 0 in all other cases. We set G = SO(E). We note that Doty, [20], also considers the algebras defined by the full orthogonal group, but we restrict ourselves to the special orthogonal group. We identify G with a subgroup of GLn (k) via our choice of basis. Let T be the maximal torus of G consisting of diagonal matrices. Then X(T ) has Z-basis 1 , . . . , m , where i takes an element of T to its (i, i) entry. We identity an m-tuple of integers λ = (λ1 , . . . , λm ) with λ1 1 +· · ·+λm m . Let Φ denote the set of roots. Then Φ consists of the elements ±i ±j , 1  i, j  m, i = j. The elements 1 , . . . , m form an orthonormal basis of a W = NG (T )/T invariant, symmetric, non-singular bilinear form ( , ) on R ⊗Z X(T ). Let Φ+ be the set of elements i − j , 1  i < j  m, and i + j , 1  i, j  m, i = j. The corresponding negative Borel subgroup consisting of the lower triangular matrices in G. For λ ∈ X(T ) we have the induced module ∇(λ) = indG B kλ . By the result of Brundan, [1, Proposition 3.3] the restriction of a rational SLn (k)-module with a good filtration to r G has a good filtration. In particular each exterior power E has a good filtration as a G-module. r For 1  r  m − 1, by character considerations (see e.g., [7, (4.1.1)]), we have E= r r ∇(1 +· · ·+r ) and since E is self dual as a G-module we have E = M (1 +· · ·+r ). m Also, by character considerations, E has a filtration with sections ∇(1, . . . , 1, −1) and ∇(1, . . . , 1, 1). Moreover, since the difference (1, . . . , 1, 1) − (1, . . . , 1, −1) = (0, . . . , 0, 2) is not a sum of positive roots, every G-module extension 0 → ∇(1, . . . , 1, −1) → Y → ∇(1, . . . , 1, 1) → 0 splits (see for example [25, II, 4.15 Lemma]) so that m

E = ∇(1, . . . , 1, −1) ⊕ ∇(1, . . . , 1, 1).

m However, E is a tilting module and it follows that M (1, . . . , 1, −1) = ∇(1, . . . , 1, −1) and M (1, . . . , 1, 1) = ∇(1, . . . , 1, 1). In particular we have m      cf M (1, . . . , 1, −1)  cf E  cf E ⊗m . Now let π be the saturation of r1 . We recall the description of π from [22, Section 1.3]. For s  0 we write Λ− (m, s) for the set of m-tuples of integers λ = (λ1 , . . . , λm−1 , λm ) with λm  0 and λ1  · · ·  λm−1  |λm | and λ1 + · · · + λm−1 + |λm | = s. Then by

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considering the weights of E ⊗r it is shown in [22, Section 1.3] that π=





 Λ+ (m, m − 2s) ∪ Λ− (m, r − 2s) .

0sr/2

We claim that     cf M (λ)  cf E ⊗r λ  for all λ ∈ π. If λ ∈ Λ+ (m, r − 2s), the module E has a highest weight λ. Moreover, λ  λ  E is a tilting module so that M (λ) is a summand of E. Hence we have cf(M (λ))  λ λ ⊗r−2s E). Moreover, E is a quotient of E so we obtain cf( λ        cf M (λ)  cf E  cf E ⊗(r−2s)  cf E ⊗r . We now show that cf(M (λ))  cf(E ⊗r ) for λ ∈ Λ− (m, r − 2s). It suffices to prove that for all r if λ ∈ Λ− (m, r) then cf(M (λ))  cf(E ⊗r ) (since cf(E ⊗(r−2s) )  cf(E ⊗r )). If this is not so let r be minimal for which the property fails. We write λ = (λ, . . . , λm ), so λ1  · · ·  λm−1  |λm |, with λm < 0. We can write λ = μ + τ with τ = (1, . . . , 1, −1). Now M (λ) is a direct summand of M (μ) ⊗ M (1, . . . , 1, −1) so that cf(M (μ))  cf(E ⊗r−m ) by induction and it suffices to prove that cf(M (1, . . . , 1, −1))  cf(E ⊗m ). This was noted above. Hence we have cf(M (λ))  cf(E ⊗r ) for all tilting modules M (λ), λ ∈ π, and hence A(π) =



    cf M (λ)  cf E ⊗r .

λ∈π

On the other hand E ⊗r is a module belonging to π so we have cf(E ⊗r )  A(π) and hence   cf E ⊗r = A(π). Thus cf(E ⊗r ) is a generalised Schur coalgebra and Sr (G) = cf(E ⊗r )∗ = S(π) is a generalised Schur algebra. 2.6. We now consider type B. We assume the characteristic of k is not 2. Let G be the simply connected, semisimple group over k of type Bm , m  2. We choose a maximal torus T , Borel subgroup B and so on as in Section 1. We choose the simple roots ˇ i+1 ) = −1 for 1  i < m − 1 and (αm−1 , α ˇ m ) = −2. (Here α1 , . . . , αm such that (αi , α α ˇ i = 2αi /(αi , αi ), for 1  i  m.) Let 1 , . . . , m ∈ X(T ) be such that the fundamental dominant weights corresponding to α1 , . . . , αm are 1 , 1 + 2 , . . . , 1 + · · · + m−1 , 12 (1 + · · · + m ). We identify λ = (λ1 , . . . , λm ) ∈ Zm with λ1 1 + · · · + λm m ∈ X(T ). The half ¯ for sum of positive roots ρ is 12 (2m − 1, . . . , 3, 1). We write E for ∇(1, 0, . . . , 0) and G

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the image of G in GL(E). The weights of E are ±i , 1  i  m, and 0. The G-module E is self dual and we may choose corresponding weight vectors basis e1 , . . . , en of E such that G preserves the bilinear form defined by (ei , ej ) = 1 if i + j = n + 1 and (ei , ej ) = 0 ¯ is the subgroup of GL(E) consisting of elements in all other cases. Then we have that G which preserve the form and have determinant 1. Let J denote the n×n matrix over k with (i, j) entry 1 if i+j = n+1 and 0 otherwise. ¯ with a group of matrices via our choice of basis. Thus G ¯ is the special We identify G   orthogonal group SO(J) = {g ∈ SLn (k) | g Jg = J} (where g denotes the transpose of the matrix g). We consider also the full orthogonal group O(J) = {g ∈ GLn (k) | g  Jg = J} and GO(J), the product of O(J) and the group of scalar matrices. Each scalar matrix acts on E ⊗r by scalar multiplication so we have     ¯ = Sr O(J) = Sr GO(J) . Sr (G) = Sr (G) By the result of Brundan, [1, Proposition 3.3] the restriction of a rational SL(E)-modur le with a good filtration to G has a good filtration. In particular an exterior power E ˜ has a good filtration as a G-module. By character considerations (see e.g., [7, (4.1.1)]), r for 1  r  m, we have E = ∇(ωr ), where ωr = 1 + · · · + r . We write also Ar (m) for the coefficient space cf(E ⊗r ) and Sr (m) for the dual algebra. Thus Sr (m) acts faithfully on E ⊗r and Sr (G) is its image in Endk (E ⊗r ). By the Remark of Section 2.2, we have Ar (m)  Ar+2 (m). We call an element λ ∈ Λ+ (m) admissible with respect to a non-negative integer r if either: (i) λ is a partition of s for some s  r with r − s even; or (ii) |λ| + 2(m − l(λ))  r. Here l(λ) denotes the length of the partition λ. We write Λ+ (m, r)ad for the set of all λ ∈ Λ+ (m) admissible with respect to r. The set of λ ∈ Λ+ (m) such that (E ⊗r : ∇(λ)) = 0 is, by character considerations, independent of the characteristic. We therefore have, as in [22, Theorem A2], by a result of Weyl, that (E ⊗r : ∇(λ)) = 0 if and only if λ ∈ Λ+ (m, r)ad . For λ ∈ Λ+ (m) we define f (λ) = |λ| + 2(m − l(λ)). We first prove the following result. Lemma 1. For λ ∈ Λ+ (m) we have cf(M (λ))  Af (λ)+1 (m). Proof. First note that the result is valid if λ = 0 or λ = ωj , for some 1  j  m. Set r = f (λ) + 1. Then (E ⊗r : ∇(λ)) = 0 by Weyl’s criterion, and so cf(∇(λ))  cf(E ⊗r ) = Ar (m). However, we have ∇(λ) = M (λ) and so cf(M (λ))  Ar (m). Now suppose for a contradiction that the lemma is false and that λ ∈ Λ+ (m) has minimal degree such that cf(M (λ))  Af (λ)+1 (m) and set r = f (λ) + 1. Let l = l(λ). If λl  2 then we may write λ = μ + ωl , with μ ∈ Λ+ (m), and f (λ) = |μ| + l + 2(m − l) =

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f (μ) + l. By the minimality assumption we have cf(M (μ))  Af (μ)+1 (m). But now, the l tilting module M (μ) ⊗ E has highest weight λ so that M (λ) is a summand and  l      l  cf M (λ)  cf M (μ) ⊗ E = cf M (μ) .cf E  ⊗l   Af (μ)+1 (m).cf E (since

l

E is a factor module of E ⊗r ) so we have   cf M (λ)  Af (μ)+1 (m)Al (m)  Af (μ)+1+l (m) = Af (λ)+1 (m).

So in fact we have λl = 1 and λ1 = λ2 = · · · = λt > λt+1  · · ·  λl , for some t E and argue as above to derive 1  t < l. We write λ = μ + ωt and consider M (μ) ⊗ a contradiction. 2 Lemma 2. For λ ∈ Λ+ (m, r)ad we have cf(M (λ))  Ar (m). λ Proof. Suppose first that r = |λ| + 2s, for some s  0. Then E is a tilting module with highest weight λ, and so contains M (λ) as a summand. Thus we have λ      cf M (λ)  cf E  cf E ⊗|λ| = A|λ| (m) and we have A|λ| (m)  Ar (m), since r − |λ| is even (see the Remark of Section 2.2). Suppose the result is false and r is minimal such that there exists some λ ∈ Λ+ (m, r)ad with cf(M (λ))  Ar (m). Then, by the above, r − |λ| is odd and f (λ)  r − 1. Thus r − 1 = f (λ) + 2s for some s and   cf M (λ)  Af (λ)+1 (m) = Ar−2s (m)  Ar (m).

2

r We set σ(m, r) = s=0 Λ+ (m, s). Then σ(m, r) is saturated in X + (T ) and so defines a generalised Schur algebra, in particular a quasi-hereditary algebra. The interesting question is whether π = Λ+ (m, r)ad satisfies the condition of the 2.1 Corollary. We consider the following hypothesis. (H) Whenever λ ∈ Λ+ (m, r)ad and μ ∈ Λ+ (m) are such that [∇(λ) : L(μ)] = 0 then μ ∈ Λ+ (m, r)ad . Proposition. The algebra Sr (G) is quasi-hereditary with respect to the labelling of weights {L(λ) | λ ∈ Λ+ (m, r)ad } and costandard modules ∇(λ) = indG B kλ providing that (H) is satisfied. Proof. Put π = Λ+ (m, r)ad . Note that the hypothesis implies that E ⊗r belongs to π. The module E ⊗r has a good filtration with sections ∇(λ), λ ∈ π = Λ+ (m, r)ad , so that

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a composition factor L(μ), μ ∈ Λ+ (m), of E ⊗r is a composition factor of ∇(λ) for some λ ∈ π and hence μ ∈ π, by the hypothesis. Also, if (H) is satisfied then the dual algebra S(π) of A(π) is quasi-hereditary and mod(S(π)) is a highest weight category with respect to the induced order on π, by 2.1  Corollary. Thus we have A(π) = λ∈π cf(M (λ)), by 1.2 Proposition. So by Lemma 2 above we have A(π)  Ar (m). Since E ⊗r belongs to π we have Ar (m) = cf(E ⊗r )  A(π) and so Ar (m) = A(π). Thus Sr (G) = Ar (m)∗ = A(π)∗ = S(π) is quasi-hereditary. 2 We check that (H) is satisfied if m  2 or r  2 by a case-by-case analysis. We have also checked various other examples not considered here but do not know whether to expect the hypothesis to hold in general. Case 1. r = 1. We have Λ+ (m, 1)ad = {1}. Moreover, ∇(1) = E is simple so the condition is satisfied and S1 (m) is quasi-hereditary. Case 2. m = 1. We may assume r  2 by the case just considered. Hence 0 ∈ Λ+ (1, r)ad and indeed Λ (1, r)ad = σ(1, r), which is saturated and Sr (1) is a generalised Schur algebra. +

Case 3. r = 2. We may assume that m  2 by Case 2. Let π = Λ+ (m, 2)ad = {2, (1, 1), 0}. To show that π satisfies the conditions of the lemma it suffices to show that [∇(2) : L(1)] = 0. We note that the calculation of [∇(2) : L(1, 1)] is reduced to an SL2 (k) result, by [25, II, 5.20, (2)], since (2, 0) − (1, 1) = (1, −1) is a simple root, and therefore (since p > 2) we have [∇(2) : L(1, 1)] = 0. Hence ∇(2)/L(2) has possible composition factors L(1), L(0). However L(1) = E, L(0) = k and we have Ext1G (E, k) = Ext1G (k, E) = H 1 (G, E) = 0 (since E = ∇(1)). Hence ∇(2)/L(2) is semisimple and [∇(2) : L(1)] = dim HomG (∇(2), E). However, ∇(2) is a quotient of E ⊗ E and we have HomG (E ⊗ E, E) = H 0 (G, E ⊗ E ⊗ E) = 0, since (by character considerations) ∇(0) = k does not appear as a section in a ∇-filtration of E ⊗ E ⊗ E, see e.g., [25, II, 4.16 Proposition]. Case 4. m = 2. We may assume that r > 2, by Case 3. Then Λ+ (2, r)ad = σ(2, r)\{(r−1, 0)}. It suffices to prove that [∇(λ) : L(r − 1, 0)] = 0, for λ ∈ Λ+ (2, r)ad . Since [∇(λ) : L(r − 1, 0)] = 0 implies λ  (r − 1, 0) it suffices to prove that [∇(r, 0) : L(r − 1, 0)] = 0.

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Recall that a Weyl module Δ(λ) has a filtration Δ(λ) = Δ0 (λ) ⊃ Δ1 (λ) ⊃ Δ2 (λ) ⊃ · · · with Δ0 (λ)/Δ1 (λ) = L(λ) and that we have Jantzen’s sum formula 

ch Δi (λ) = JSp (λ)

i1

where 

JSp (λ) =



νp (jp)χ(sα,jp · λ).

α>0 0
Here νp is the p-adic valuation on positive integers and sα,jp is the affine reflection defined by α and jp. (For further details see, [25, II, Section 8.19].) If (λ + ρ, α ˇ ) = jp + a then sα,jp · λ = λ − aα. Each χ(sα,jp · λ) is either 0 or ±χ(μ) for some dominant weight μ. Hence we have 

JSp (λ) =

μ∈Λ+ (m)

aμ χ(μ) −



bμ χ(μ)

μ∈Λ+ (m)

where aμ is the sum of those νp (pj) for pairs (α, j) with 0 < jp < (λ+ρ, α) ˇ and χ(sα,jp ) = +χ(μ) and where bμ is the sum of those νp (pj) for pairs (α, j) with 0 < jp < (λ + ρ, α) ˇ + and χ(sα,jp ) = −χ(μ). Thus, for ξ ∈ X (T ) we have  i1

 Δi (λ) : L(ξ) =

 μ∈Λ+ (m)

  aμ Δ(μ) : L(ξ) −



  bμ Δ(μ) : L(ξ) .

(∗)

μ∈Λ+ (m)

We now take λ = (r, 0) and ξ = (r − 1, 0). A contribution to the right hand side comes from μ such that [Δ(μ) : L(ξ)] = 0, so that (r − 1, 0)  μ < (r, 0). The only possibility is μ = (r − 1, 0). So we have to investigate those pairs (α, j) with χ(sα,jp · λ) = ±χ(r − 1, 0). We have Φ+ = {1 , 2 , 1 − 2 , 1 + 2 }. We consider the relevant contributions made to JSp (λ) made by α ∈ Φ+ in order or increasing difficulty. (i) We first consider the case α = 2 . Then (λ + ρ, α ˇ ) = 3 and there is no contribution. (ii) Now consider the case α = 1 +2 . Then we have (λ+ρ, α ˇ ) = r+2. Suppose we have r +2 = jp+a, 0 < jp < r +2. Then we have a term νp (jp)χ(λ−aα) = νp (jp)χ(r −a, −a) appearing in the Jantzen sum. We have χ(r − a, −a) = −χ(r − a, a − 1). If r − a  a − 2 then χ(r −a, a−1) is a Weyl character or zero. So we get a relevant contribution precisely when a = 1, so that p divides r+1 and the contributions is precisely −νp (r+1)χ(r−1, 0). Now suppose r − a < a − 2. i.e., r < 2a − 2. Then −χ(r − a, a − 1) = χ(a − 2, r − a + 1). This is a Weyl character that makes no contribution to the multiplicity calculation. (iii) Now consider the case α = 1 − 2 . Then (λ + ρ, α) ˇ = r + 1. Suppose we have jp + a = r + 1, 0 < jp < r + 1. Then we have a term νp (jp)χ(r − a, a) appearing in the Jantzen sum. If r − a  a − 1 then χ(r − a, a) is a Weyl character or zero and does

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not contribute to the calculation. So we assume r − a < a − 1, i.e., r + 1 < 2a. Then νp (jp)χ(r −a, a) = −νp (jp)χ(a−1, r −a+1). We cannot have (a−1, r −a+1) = (r −1, 0) so this makes no contribution to the calculation. (iv) Finally we consider the case α = 1 . We have (λ+ρ, α) ˇ = 2r +3. Suppose we have jp+a = 2r+3, with 0 < jp < 2r+3. Then we have a term νp (pj)(r−a, 0) appearing in the Jantzen sum. We get a contribution νp (pj)χ(r−1, 0) in case a = 1, so 2r+3 ≡ 1 modulo p, i.e. p divides r+1. So the precise contribution is νp (2r+2)χ(r−1, 0) = νp (r+1)χ(r−1, 0). For a further relevant contribution we must have r + 1 < a < 2r + 3. Then we have χ(r − a, 0) = −χ(a − r − 3, 0). If a = r + 2 this is zero. However (a − r − 3, 0) = (r − 1, 0) for a = 2r + 2. This is impossible since then jp = 1. Collecting together the contributions from the second and fourth cases considered above we have     Δ(r, 0) : L(r − 1, 0) = −νp (r + 1) χ(r − 1, 0) : L(r − 1, 0)   + νp (r + 1) χ(r − 1, 0) : L(r − 1, 0) = 0. Thus the hypothesis (H) is satisfied and Sr (G) is quasi-hereditary if G has semisimple rank 2. 3. The defining ideal of a generalised Schur algebra 3.1. We initially restrict ourselves to the representation theory of a hyperalgebra over an arbitrary field, defined by a complex semisimple Lie algebra, for simplicity of exposition. Later we shall indicate the modifications necessary to cover some variations. Let g be a finite dimensional, complex, semisimple Lie algebra with Cartan subalgebra h and Weyl group W . Let Φ+ (resp. Φ− ) be a system of positive roots (resp. negative roots). We write X for the set of integral weights and X + for the set of dominant weights. We fix a Chevalley basis {Xα | α ∈ Φ} ∪ {Hi | 1  i  l} (such that H1 , . . . , Hl ∈ h). Let   n+ = α∈Φ+ CXα and n− = α∈Φ− CXα . Let UC denote the enveloping algebra of g. We define the elements Xα,r =

Xαr r!

and Hi,b =

Hi (Hi − 1) . . . (Hi − b + 1) b!

of UC , for α ∈ Φ, r  0 and 1  i  n, b  0. The elements Xα,r , α ∈ Φ, r  0, generate a Z-form UZ of UC , known as the Kostant Z-form. Let k be a field and let Uk (g) = k ⊗Z UZ be the hyperalgebra. For α ∈ Φ, a  0 we write xα,r for the element 1 ⊗ Xα,r of Uk (g) and for 1  i  l and b  0, write hi,b for the element 1 ⊗ Hi,b of Uk (g). We shall write Uk (h) for the subalgebra of Uk (g) generated by the elements hi,b , 1  i  l, b  0. We write Uk (n+ ) (resp. Uk (n− )) for the subalgebra of Uk (g) generated by all elements xα,r with α > 0 (resp. α < 0). Then multiplication Uk (n+ ) ⊗ Uk (h) ⊗ Uk (n− ) → Uk (g) is an isomorphism of k-vector spaces.

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For an integral weight λ ∈ X we also denote by λ the k-algebra homomorphism from  i ) 1k . For λ ∈ X we set Mλ = Ker(λ), a codimension one Uk (h) to k taking hi,b to λ(H b ideal of Uk (h). If V is a Uk (g)-module and λ ∈ X then we have the weight space 

V λ = v ∈ V hv = λ(h)v for all h ∈ Uk (h) . A finite dimensional Uk (g)-module V is a direct sum of its (integral) weight spaces, i.e., we have  V = V λ. λ∈X

For each λ ∈ X + there is a finite dimensional Uk (g)-module L(λ) which has highest weight λ. Moreover, λ occurs with multiplicity one as a weight in L(λ) and {L(λ) | λ ∈ X + } is a complete set of pairwise non-isomorphic irreducible finite dimensional Uk (g)-modules. For λ ∈ X + we write wtk (λ) for the subset of X consisting of all weights of L(λ). We recall some elementary general properties of ideals of codimension 1. Let A be a k-algebra and let {Mλ | λ ∈ F } be a finite set of codimension 1 ideals, with Mλ = Mμ  for λ = μ. We put kλ = A/Mλ , for λ ∈ F . We put MF = λ∈F Mλ . Then A/MF  embeds in λ∈F A/Mλ and hence dim A/MF  |F |. Moreover, we have MF  Mλ  A so that kλ = A/Mλ is a composition factor of A/MF , for each λ ∈ F . Hence the dimension of A/MF is at least |F | and therefore precisely |F |. Thus the natural map   A/MF → λ∈F A/Mλ is an isomorphism, in particular we have A/MF ∼ = λ∈F kλ , as A-modules. One deduces that an A-module V is a direct sum of copies of kλ , λ ∈ F , if and only if MF V = 0. We now apply this to the case A = Uk (h) and, as above, for λ ∈ X, write Mλ for the kernel of the corresponding homomorphism Uk (h) → k. For a finite (not necessarily saturated) subset π of X + we define Ik (π) to be the ideal of Uk (g) consisting of the elements which act as 0 on every finite dimensional Uk (g)-module whose composition factors belong to {L(λ) | λ ∈ π}. Remark. We note that Ik (π) has finite codimension in Uk (g). For π saturated this is true by [9, 3.2, (1)]. In general we may choose σ a finite saturated subset of X + containing π. Then we have Ik (σ)  Ik (π) and since Uk (g)/Ik (σ) is finite dimensional, so is Uk (g)/Ik (π). Proposition. Let F be a finite subset of X. The ideal of Uk (g) generated by the ideal MF of Uk (h) is equal to Ik (πF ), where πF is the set of λ ∈ X + such that wtk (λ) ⊆ F . In particular if π is a finite set of dominant weights then Ik (π) is generated by its intersection with Uk (h). Proof. For an arbitrary Uk (h)-module Y and a set E of integral weights we write Y E for  λ wt for Y X , the sum of all integral weight spaces. Note that, since λ∈E Y . We write Y

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Y wt is a direct sum of one dimensional Uk (h)-modules it is locally finite dimensional, i.e., it has the property that every finite dimensional subspace of Y wt is contained in a finite dimensional Uk (h)-submodule. Note too that if Y is a Uk (g)-module, α is a root and r is a non-negative integer then for any weight λ we have xα,r Y λ  Y λ+rα . Thus Y wt is a Uk (g)-submodule of Y . We write JF for the ideal of Uk (g) generated by MF . Let V = Uk (g)/JF and regard V as a left Uk (g)-module. Then V is annihilated by MF and hence is the sum of weights spaces V λ with λ ∈ F . Now, for λ ∈ F , α a root and r sufficiently large we have λ+rα ∈ /F (since F is finite) and hence we must have xα,r V  V λ+rα = 0, i.e. xα,r (Uk (g)/JF ) = 0. i.e., xα,r Uk (g)  JF , i.e., xα,r ∈ JF . Now if Z is a finite dimensional subspace of Uk (g)/JF then Z is contained in a finite dimensional Uk (h)-submodule of Uk (g)/JF (since Uk (g)/JF is a direct sum of its weight spaces) and hence there exists an integer N such that xα,r Z = 0 for all roots α and all r  N . It follows that Uk (h)Uk (n− )Z is finite dimensional and hence Uk (g)Z = Uk (n+ )Uk (h)Uk (n− )Z is finite dimensional. But Uk /(g)JF is generated by 1 + JF and hence Uk (g)/JF is finite dimensional, i.e., JF has finite codimension. Let L(λ) be a composition factor of V . Then since MF V = 0 we also have MF L(λ) = 0 and hence MF annihilates every weight space of L(λ), i.e. every weight of L(λ) belongs to F . Thus all composition factors of V belong to {L(λ) | λ ∈ πF }. Hence Uk (g)/J is annihilated by Ik (πF ) and so Ik (πF )  JF . We now show the reverse inclusion. The quotient Uk (g)/Ik (πF ) is a finite dimensional Uk (g)-module all of whose composition factors belong to {L(λ) | λ ∈ πF }. Hence Uk (g)/Ik (πF ) is the direct sum of its weight spaces and each weight is a weight of L(λ) for some λ ∈ πF and hence belongs to F . Thus we have MF (Uk (g)/Ik (πF )) = 0, i.e., MF Uk (g)  Ik (πF ) and hence JF  Ik (πF ) and we have JF = Ik (πF ), as required. 2 Remark. Let Gk be the corresponding simply connected, Chevalley group scheme over k. We identify Uk (g) with the algebra of distributions of Gk . Each finite dimensional Uk (g) module is naturally a Gk -module (see for example [25, Section II, 1.20]). For π a finite set of dominant weights let Ak (π) be the smallest subcoalgebra of k[Gk ] such that every Gk -module with composition factors in {L(λ) | λ ∈ π} has coefficient space in Ak (π), as in [9]. Then Ak (π) is finite dimensional with dual algebra Sk (π). This is a generalised Schur algebra as discussed in [9] if π is saturated. There is a natural surjection Uk (g) → Sk (π) with kernel Ik (π) (see [9, Section 3] for further details). Hence the above Proposition gives, at least in principle, a description of the algebra Sk (π) by generators and relations (modulo a description of Uk (g) by generators and relations). 3.2. We now consider validity over Z of results of this kind. We will use the following elementary remark. (1) Let M be an abelian group and with submodules A  B. Suppose that M/B is free and B/A is finitely generated. If A + pM = B + pM for all primes p then A = B.

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Proof. If not then we can choose a subgroup A  X  B with |B/X| = p for some prime p. Thus pB  X. We choose a complement F of B in M . Thus we have B + pM = A + pM = X + pM i.e., B + pF = X + pB + pF = X + pF and so B = X + (B ∩ pF ) = X a contradiction. 2 For a UZ (h)-module V we write Vfin for sum of all submodules that are finitely generated over Z. Recall that, for 1  i  l, α ∈ Φ, we have the formula 



X r Hi − rα(Hi ) Hi Xαr = α b r! r! b

for b, r  0. Let V be a UZ -module and let M be a UZ (h)-submodule finitely generated as a Z-module. It follows from the above formula that Xα,r M is a UZ (h)-submodule, for α ∈ Φ, r  0. Hence Vfin is a UZ -submodule of V . Now let I be any ideal of UZ such that UZ /I is finitely generated, as a Z-module. Let J = UZ (h) ∩ I and let I  be the ideal of UZ generated by J. Note that UZ (h)/J is isomorphic to (I +UZ (h))/I as a Z-modules. Hence UZ (h)/J is a finitely generated torsion free Z-module. Now J annihilates UZ /I  and so the image of UZ (g) in UZ /I  is a finitely generated Z-module. Hence (1 + I  )/I  lies in a UZ (h)-submodule, finitely generated over Z. Hence (1 + I  )/I  ∈ Vfin , where V = UZ /I  . So Vfin is a UZ -submodule of V containing the generator (1 + I  )/I  . Hence Vfin = V , i.e., V = UZ /I  is finitely generated as a Z-module. We now let π be a finite saturated set of dominant weights. We write IZ (π) for the ideal consisting of the elements of UZ that annihilate all UZ -modules finite and free over  Z with all weights in W π. Then UZ /IZ (π) is free over Z of rank λ∈π (dimC ΔC (λ))2 , by [13, Lemma 1.2c]. We put JZ (π) = UZ (h) ∩ IZ (π) and let I  (π) be the ideal of UZ  generated by JZ (π). Then UZ (h)/JZ (π) = λ∈W π Zλ . (Here Zλ denotes UZ (h)-module which is free of rank one as a Z-module and on which UZ acts via λ, for λ ∈ X.) We now consider the corresponding situation over the field Fp of p elements, where p is a prime. We write IFp (π) for the ideal of UFp consisting of all elements that vanish on all finite dimensional UFp -modules belonging to π. Then IFp (π) = Fp ⊗Z IZ (π). We write JFp (π) for UFp (h) ∩ IFp (π), the annihilator in UFp (h) of UFp /IFp (π). As a UFp (h)-module  UFp (h)/JFp (π) is λ∈W π Fp,λ . It follows that JFp (h) = Fp ⊗Z JZ (π). Now by 3.1 Proposition we have

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IFp (π) = UFp .JFp (π).UFp hence we have 

   IZ (π) + pUZ (π) /pUZ = UZ .JZ (π).UZ + pUZ /pUZ

so that IZ (π) + pUZ (π) = I  (π) + pUZ for all primes p. By (1) above we have IZ (π) = I  (π). We have now proved the following result. Theorem. Let π be a finite saturated subset of X + . The ideal IZ (π) is generated by its intersection with UZ (h). Recall that there is natural ring epimorphism UZ → SZ (π) with kernel IZ (π), see e.g. [9, Section 3], so the Theorem gives, at least in principle, a description of the integral Schur algebra by generators and relations (modulo a description of UZ by generators and relations). 3.3. Finally we consider the case of generalised Schur algebras defined by general linear groups and Lie algebras. Let g denote the complex Lie algebra gln (C), n  2. Let h denote the Lie subalgebra consisting of all diagonal matrices. For 1  i  n let i ∈ h∗ = HomC (h, C) be the linear functional taking an element of h to its (i, i) entry. We let E denote the R span of 1 , . . . , n and regard E as a Euclidean form via the real bilinear form for which 1 , . . . , n is an orthonormal basis. We write Φ for the set of all i − j , i = i, and Φ+ for the set of all i − j with i < j. Let E0 be the R-span of Φ. Then (E0 , Φ) is a root system with system of positive roots Φ+ . The symmetric group W = Sym(n) acts on E by permuting 1 , . . . , n and we identify W with the Weyl group by restricting the action to E0 . We identify X(n) = Zn with a subgroup of h∗ by identifying λ = (λ1 , . . . , λn ) ∈ X(n) with λ1 1 + · · · + λn n . For α = i − j ∈ Φ we write Xα for the element of g which has entry 1 in the (i, j) position and entry 0 in all other positions. For 1  i  n we write Hi for the element of g which has entry 1 in the (i, i) position and all other entries 0. Then g has a Z-form gZ = gln (Z) with Z-basis {Hi | 1  i  n} ∪ {Xα | α ∈ Φ}. Let UC denote the universal enveloping algebra of g. We denote by UZ the subring of UC generated by all Xα,r =

Xαr r!

and Hi,b =

Hi (Hi − 1) · · · (Hi − b + 1) b!

for all α ∈ Φ, r  0, and all, 1  i  n, b  0. Then UZ is a Hopf Z-form of UC . We write UZ (h) for the subring generated by all Hi,b . Then UZ (h) is a summand of UZ as

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a Z-module and a Hopf Z-form of the enveloping algebra UC (h) of h. For a field k, we write Uk and Uk (h) for the Hopf algebras k ⊗Z UZ and k ⊗Z UZ (h) obtained by base change. We define elements xα,r = 1 ⊗ Xα,r

and hi,b = 1 ⊗

 Hi b

for α ∈ Φ, r  0 and 1  i  n, b  0. For λ = (λ1 , . . . , λn ) ∈ X(n) we have the ring homomorphism (also denoted λ)   from UZ (h) to Z taking Hi,b to λbi . We also have the k-algebra homomorphism (also   denoted λ) from Uk (h) to k taking hi,b to λbi 1k . In this way we identify X(n) with a group of multiplicative characters of UZ (h) and of Uk (h). In all contexts we shall refer to an element of X(n) as an integral weight. For a Uk (h)-module V and λ ∈ X(n) we have the weight space 

V λ = v ∈ V hv = λ(h) for all h ∈ Uk (h) . As usual the elements λ ∈ X(n) such that V λ = 0 will be called the weights of V . We shall say that a Uk (h)-module V is admissible if V =



V λ.

λ∈X(n)

We have the usual terminology of dominance order, dominant weights etc. for X(n) (as in Section 1). Let X + (n) denote the set of dominant weights. For λ ∈ X + (n) there is a unique, up to isomorphism finite dimensional irreducible module L(λ) with highest weight λ and {L(λ) | λ ∈ X + (n)} is a complete set of pairwise non-isomorphic finite dimensional simple modules. Let π be a finite (not necessarily saturated) set of dominant weight. Let Ik (π) denote the ideal of Uk consisting of the elements that annihilate all finite dimensional admissible Uk -modules whose composition factors come from the set {L(λ) | λ ∈ π}. Then we have, by the argument of 3.1 Proposition: Proposition 1. Ik (π) is generated by its intersection with Uk (h). We now turn to the integral case. We now suppose that π is saturated. For a UZ (h)-module V and λ ∈ X(n) we define  Vλ =

  λi v for all 1  i  n, b  0 . v ∈ V Hi,b v = b

We say a UZ (h)-module V is admissible if it is finitely generated and free as a Z-module and V =

 λ∈X(n)

V λ.

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The elements λ ∈ X(n) such that V λ = 0 are as usual called the weights of V . We now take π to be a finite saturated set of dominant weights. We shall say that an admissible module V belongs to π if all weights of V belong to W π. We denote by IZ (π) the ideal of UZ consisting of those elements that annihilate all admissible UZ -modules belonging to π. Then arguing as in Section 3.2 we have Proposition 2. For a finite saturated set of dominant weights π, the ideal IZ (π) is generated by its intersection with UZ (h). 3.4. We now relate these results to the Schur algebras of general linear group schemes. Let Ak (n) = k[x11 , . . . , xnn ], the free commutative polynomial algebra in variables xij , 1  i, j  n. Let d = det(xij ) ∈ Ak (n) be the determinant in the variables xij . Let Gk be the general linear group scheme of degree n over k. Thus the coordinate algebra k[Gk ] is the localisation of Ak (n) at d, comultiplication is the k-algebra map δ : k[Gk ] → n k[Gk ] ⊗ k[Gk ] taking xij to r=1 xir ⊗ xrj and the augmentation map k : k[Gk ] → k is the k-algebra map taking xij to the Kronecker delta δij . We denote by Tk the group scheme of Gk whose defining ideal is generated by all xij with i = j. The coordinate −1 algebra k[Tk ] is the Laurent polynomial algebra k[t1 , t−1 1 , . . . , tn , tn ], where ti is the image of xii under the natural map k[Gk ] → k[Tk ]. For λ = (λ1 , . . . , λn ) ∈ X(T ) we set tλ = tλ1 1 . . . tλnn . For a left Gk -module V , i.e., a right k[Tk ]-comodule V , and λ ∈ X(n) we set 

V λ = v ∈ V τ (v) = v ⊗ tλ where τ : V → V ⊗ k[Tk ] is the comodule structure map. Then we have V =



V λ.

λ∈X(n)

An element λ ∈ X(n) such that V λ = 0 is called a weight of V . For each λ ∈ X + (n) there is a unique (up to isomorphic) simple module Lk (λ) with highest weight λ. Moreover, {L(λ) | λ ∈ X + (n)} is a complete set of pairwise non-isomorphic irreducible Gk -modules. We now take k = C. We identify the Lie algebra of GC with g = gln (C) in the usual way. The coalgebra structure on C[GC ] gives rise to an algebra structure on C[GC ]∗ and inclusion gC to HomC (C[GC ], C) extends to an algebra monomorphism. In this way we identity the enveloping algebra UC of gC with the algebra of distributions of gC . We write GZ for the general linear group scheme over Z. The coordinate ring Z[GC ] is the subring of C[GC ] generated by the xij and 1/d. Moreover, UC has Hopf Z-form UZ , the subring generated by elements Hi,b and Xα,r , with 1  b  n, b  0, α ∈ Φ, r  0. For all γ ∈ UZ we have γ(Z[GZ ])  Z and indeed the induced map UZ → HomZ (Z[GZ ], Z) identifies UZ with the Z-algebra of distributions of GZ . We obtain by base change an identification of Uk with the algebra of distributions of k[Gk ] = k ⊗Z Z[GZ ].

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Now let π be a finite subset of X + (n). The smallest subcoalgebra of k[Gk ] such that the coefficient space of every Gk -modules all of whose composition factors belong to π will be denoted Ak (π). Then Ak (π) is finite dimensional and we denote its dual algebra by Sk (π). Identifying Uk with the algebra of distributions, as above, we have the natural surjection Uk → Sk (π) with kernel Ik (π). Thus 3.3, Proposition 1 provides a description of Sk (π) by generators and relations (modulo a description of Uk by generators and relations). Now suppose π is saturated. We write AZ (π) for AC (π) ∩ Z[GZ ]. Then AZ (π) is finitely generated and free as a Z-module. Let SZ (π) denote the dual algebra. The natural map UZ → AZ (π) induces an isomorphism UZ /I(π) → SZ (π). Thus Proposition 2 can be used to provide a description of the integral Schur algebra SZ (π) by generators and relations. Remark. These observations apply in particular to the ordinary and rational Schur algebras, choosing π = Λ+ (n, r) and π = Λ+ (n; r, s). We explore this in the next section, but only over fields of characteristic 0. 4. The defining ideal for certain generalised Schur algebras 4.1. As a first application we deduce the presentation given by Doty and Giaquinto, (in the classical case) [21, Theorem 2.1] for the Schur algebra S(n, r). So we take k to be a field of characteristic 0. We fix π = Λ+ (n, r). Then we have a natural isomorphism Uk /I(π) → S(n, r) and the presentation of Doty and Giaquinto is equivalent to the statement that I(π) is generated by the elements Hi (Hi − 1) . . . (Hi − r),

1  i  n,

and

H1 + · · · + Hn − r.

(†)

So let I0 denote the ideal of Uk (h) generated by these elements and let I denote the ideal of Uk generated by these elements. We note that Uk (h)/I0 is finite dimensional, since it is spanned by the images of the elements H1a1 . . . Hnan , 0  a1 , . . . , an  r. Consider the quotient Uk (h)-module V = Uk (h)/I0 . Since Hi (Hi − 1) . . . (Hi − r) acts as zero on V we r have V = t=0 V (i)t , where V (i)t = {v ∈ V | Hi v = tv}, for 1  i  n. The elements H1 , . . . , Hn commute so we have V =



Vλ

λ∈F0

where F0 is the subset of X consisting of weights λ = (λ1 , . . . , λn ) with 0  λi  r, for 1  i  n. For λ = (λ1 , . . . , λn ) ∈ F0 and v ∈ V λ we have   (H1 + · · · + Hn )v = λ(H1 ) + · · · + λ(Hn ) v = (λ1 + · · · + λn )v

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and since H1 + · · · + Hn acts as multiplication by r we must have λ1 + · · · + λn = r whenever V λ = 0. Hence we have V =VF where F = Λ(n, r). Thus we have MF Uk (h)  I0 , i.e. MF  I0 . However, one sees explicitly that if H is any of the generators of I0 above and λ ∈ Λ(n, r) we have λ(H) = 0. Hence we have I0  MF and therefore I0 = MF . Now by the Proposition 1 of Section 3.3 and Section 3.4 we have that I0 generates I(π), where π = Λ+ (n, r). 4.2. We next consider the algebra defined by generators and relations by Dipper and Doty, [4, Section 7.3]. We translate back from the generators Hi given there to the “original” generators Hi , 1  i  n. The algebra is then given by generators ei , fi , Hi , with 1  i  n. The relations (a)–(e) then define the enveloping algebra Uk (g) of g = gln (k). The algebra defined by generators and relations by Dipper and Doty is then the enveloping algebra Uk (g) modulo the ideal I, say, generated by the elements Hi (Hi − 1) . . . (Hi − r − s), 1  i  s,   H1 + · · · + Hn − r + (n − 1)s .

and (†)

Once again we consider the ideal I0 of Uk (h) generated by the elements above. We note that Uk (h)/I0 is finite dimensional, since it is spanned by the elements H1a1 . . . Hnan , with 0  a1 , . . . , an  r + s − 1. Moreover, each Hi acts diagonalisably on V = Uk (h)/I0 with eigenvalues in {0, 1, . . . , r + s}. Since the elements H1 , . . . , Hn commute, we have a decomposition V =



Vλ

λ∈F0

where F0 is the set of weights λ = (λ1 , . . . , λn ) with 0  λ1 , . . . , λn  r + s. Moreover, since H1 + · · · + Hn acts as r + (n − 1)s, we must have V =VF where F is the set of all λ = (λ1 , . . . , λn ) ∈ Λ(n, r + (n − 1)s) such that 0  λ1 , . . . , λn  r + s. Thus we have MF Uk (h)  I0 , i.e., MF  I0 . However, one sees explicitly that if H is any of the generators of I0 given above and λ ∈ F , we have λ(H) = 0. Hence we have I0  MF and hence I0 = MF . Now, taking π = Λ+ (n) ∩ F , we have, by Proposition 1 of Section 3.3 and Section 3.4, that I0 generates I(π). Hence the algebra defined by Dipper and Doty is the generalised Schur algebra S(π).

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However, we note that this is not in general isomorphic to the rational Schur algebra S(n; r, s), as claimed by Dipper and Doty. For example we take n = 4, r = s = 1 so that r + (n − 1)s = 4. Then the set π above consists of the partitions (2, 2, 0, 0), (2, 1, 1, 0), (1, 1, 1, 1) and the set Λ+ (n; r, s) consists of the weights (1, 0, 0, −1), (0, 0, 0, 0). The algebra S(π), defined above thus has three classes of simple modules and the algebra S(n; r, s) has two, in particular these algebras are not isomorphic. The source of the error seems to be the claim made in the paragraph before Theorem 5.4 in [4] that V is a highest weight module for S(π) if and only if V ⊗ D⊗−s is a highest weight module for S(n; r, s) (where D is the determinant module and D⊗−s is the dual of the s-fold tensor product of D). In the above example we see that this is not true if V has highest weight (2, 2, 0, 0). We show however that at least the procedure of Dipper and Doty is valid in case n = 2 or 3. We write π(n, r; s) for the set of dominant polynomial weights μ = (μ1 , . . . , μn ) such that |μ| = r+(n−1)s and μ1  r+s. For λ ∈ Λ+ (n; r, s) it is clear that λ+sω ∈ π(n; r, s), where ω = (1, 1, . . . , 1), and so we have an injective map φ : Λ+ (n; r, s) → π(n; r, s), defined by φ(λ) = λ + sω. We suppose first that n = 2. Then π(2; r, s) is the set of 2-part partitions μ = (a, b) such that a + b = r + s, i.e., π(2; r, s) = Λ+ (2; r + s). Now suppose that μ = (a, b) ∈ Λ+ (2; r+s) and put λ = μ−sω = (a−s, b−s). If a < s then λ+ = (0, 0), λ− = (a−s, b−s) and r−|λ+ | = r and s−|λ− | = s−(2s−a−b) = −s+(r+s) = r and so μ−sω ∈ Λ+ (2; r, s). If a  s, b  s then λ+ = (a − s, 0) and λ− = (0, b − s) and r − |λ+ | = r − (a − s) = b and s − |λ− | = s − (s − b) = b so that λ ∈ Λ+ (2; r, s). If b  s then λ+ = λ and λ− = (0, 0) so that r − |λ+ | = r − (a + b − 2s) = r − (r − s) = s and s − |λ− | = s so that λ ∈ Λ+ (n, r). In all cases we have λ = μ − sω ∈ Λ+ (2; r, s). Thus φ : Λ+ (2; r, s) → π(n; r, s) is onto and hence a bijection. Now suppose that n = 3. We consider μ = (a, b, c) ∈ π(3; r, s). Thus a  b  c  0, a+b+c = r +2s and a  r +s. We show that λ = μ−sω = (a−s, b−s, c−s) ∈ Λ+ (3; r, s) and hence φ is onto. Case 1. a  s. Then λ+ = (0, 0, 0), λ− = (a − s, b − s, c − s) and r − |λ+ | = s − |λ− | = r. Case 2. b  s < a. Then λ+ = (a − s, 0, 0), λ− = (0, b − s, c − s) and r − |λ+ | = s − |λ− | = r − a + s. Case 3. c  s < b. Then λ+ = (a−s, b−s, 0), λ− = (0, 0, c−s) and r−|λ+ | = s−|λ− | = c. Case 4. s < c. Then λ+ = (a − s, b − s, c − s), λ− = (0, 0, 0) and r − |λ+ | = s − |λ− | = s. Thus in all cases we have λ = μ − sω ∈ Λ+ (n; r, s) and hence μ = φ(λ) and the map φ : Λ+ (n; r, s) → π(n; r, s) is onto, and hence a bijection.

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