Stephen Donkin

Skew modules for reductive groups

Skew modules for symmetric groups have been investigated by James and Peel [13], Farahat and Peel [9], and Clausen [S]. More recently, skew modules for general linear groups have been investigated by Akin, Buchsbaum, Weyman, [3], [l], [2], and Kouwenhoven [ 161. Our aim in this paper is to set up, in a rather general context, a theory of skew modules for reductive algebraic groups. For a partition 1, of it (James, Peel [ 13]), obtain a filtration of the Specht module Sj,, viewed as a module for the Young subgroup corresponding to a decomposition n = a + b of n. The successive quotients in the filtration have the form S, H Sj,,,p (Kronecker product), where p runs over the partitions of a such that %\p is a skew partition and S,,, denotes the skew Specht module. The analogue of this result is proved for general linear groups by Akin, Buchsbaum, and Weyman [3, Theorem 11.4.1 l] and Kouwenhoven [ 16, Proposition II, 2.2A]. The analogue for reductive groups is, more or less, our definition of skew modules. This filtration property is in fact a characterisation of skew modules (see 1.3(4)) and our definition for general linear groups therefore agrees with that of Akin, Buchsbaum, and Weyman [3] (and Kouwenhoven [ 161). It becomes apparent rather quickly, from our point of view, that the skew modules have a particular kind of filtration (see 2.1( 1)) which in the case of the general linear group is a filtration by Schur modules. This is a result of Kouwenhoven [ 16, Proposition 11.2.2B, a]), for general linear groups; Kouwenhoven is able to give an explicit description of such a filtration. One may then obtain, as in Kouwenhoven [ 16, Proposition II, 2.2B, b]), the result of James and Peel [ 131 on the existence of Specht series for skew modules for the symmetric groups by specializing to a certain weight space (i.e., applying the Schur functor, as in [ll, Chap. 63). We are also able to prove (see 2.3(3)) the existence of a certain kind of resolution considered by Akin and Buchsbaum (see [ 1, Sects. 83) for skew modules for general linear groups. Our main point of reference is the representation theory of reductive 465 0021-8693/88

Finite resolutions of modules for reductive algebraic groups

3.00

On Tilting Modules and Invariants for Algebraic Groups

of a suitable G&-module M by modules which are required to be direct sums of tensor products of exterior powers of the natural representation. In particular, in [Z, 31, for partitions 1 and p, resolutions are constructed for the modules L,(F) and L,,,(F) (the Schur functor corresponding to 1 and the skew Schur functor corresponding to (A, p), evaluated at F), where F is a free module over a field or the integers. The purpose of this paper is to characterise the GL,-modules which admit such a resolution as those modules which have a filtration in which each successive quotient has the form L,(F) for some partition p. By contrast with [2,3] we do not produce resolutions explicitly. Our methods are independent of those of Akin and Buchsbaum (we give a new proof of their result on the existence of a resolution for L,(F)) and are algebraic group theoretic in nature. We have therefore cast our main result (the theorem of Section 1) as a statement about resolutions for reductive algebraic groups over an algebraically closed field. One may frequently wish to work over a more general coefficient ring (see [l-3]) and so we treat in some detail Chevalley group schemes and general linear group schemes over a principal ideal domain (2.6 and 2.7), giving the appropriate versions of our main result and relating it to the work of Akin, Buchsbaum and Weyman.

Representations of Hecke Algebras and Characters of Symmetric Groups

The theme of this article is the calculation and description of generators of polynomial invariants for groups actions by means of trace functions. In this section we give our main applications of this point of view. In the second section we discuss the general framework and also the beautiful theory of conjugacy classes in algebraic groups due to Steinberg, where trace functions play an important role. This serves as as model for our approach in similar situations. In the third section we describe various results on tilting modules and saturated subgroups of algebraic groups. In Section 4 we explain how part of Steinberg’s set-up may be adapted to the action by conjugation of a closed subgroup H of G, in the framework of “group pairs”. The main result here is that the class functions relative to H are trace functions determined by tilting modules. We conclude with some appendices. In the main body of the text. we have omitted all but the simplest proofs. Instead references to the literature are given for proofs where appropriate and to the Appendices (which are of a somewhat technical nature) where no reference is available.

Representations of symplectic groups and the symplectic tableaux of R. C. King

We consider the representation theory of a Hecke algebra H(n) of type A at a primitive lth root of unity, where l is a (not necessarily prime) positive integer. We assign a “Brauer character” to an H(n)-module. This is a class function on the set of l-regular elements of the symmetric group S n . We prove an analogue of Brauer’s orthogonality relations. As an application of the assignment of characters we show that the determinant of the Cartan matrix divides a power of l. A precise formula for this determinant has recently been obtained by Brundan and Kleshchev. They show in particular that this is a power of l, verifying a conjecture of A. Mathas.

On conjugating representations and adjoint representations of semisimple groups

It was shown by R.C. King that symplectic standard tableaux can be used to calculate the dimensions of the weight spaces of an irreducible module for the syrnplectic group . In a recent paper. A. Berele gave a basis of this module, labelled by the symplectic standard tableaux of R. C King. The present paper gives an account of this labelling which is based on the theory of bideterminants and is valid for symplectic groups in arbitrary characteristic.

On the Existence of Auslander–Reiten Sequences of Group Representations II

where Ea is a direct sum of irreducible G-modules of highest weight 2. Here we consider the conjugating representation in arbitrary characteristic and prove the appropriate version of Richardsons Theorem (under small, and almost certainly unnecessary, characteristic/root-system restrictions). One can no longer decompose A (G) into homogeneous components but instead we prove the existence of an ascending

On Schur algebras and related algebras V: some quasi-hereditary algebras of finite type

This is the second part of our study of the existence ofAuslander–Reiten sequences of group representations. In Part I weconsidered representations of group schemes in characteristic 0; in thispart we consider representations of group schemes in characteristicp; and in Part III we give applications to representations ofabstract groups and Lie algebras.

Homological properties of quantised Borel-Schur algebras and resolutions of quantised Weyl modules

Abstract In a recent paper, Erdmann determined which of the Schur algebras S ( n , r ) have finite representation type and described the finite type Schur algebras up to Morita equivalence. The present paper grew out of a desire to see Erdmanns results in the more general context of algebras which are quasi-hereditary in the sense of Cline et al. (1988). Weconsider here the class of quasi-hereditary algebras which have a duality fixing simples. This class includes the “generalized Schur algebras” defined and studied by the first author, and the Schur algebras themselves in particular. In the first part we describe the possible Morita types of the quasi-hereditary algebras of finite representation type over an algebraically closed field with duality fixing simples. This is then applied, in the second part, to give the block theoretic refinement of Erdmanns results.

A note on the characters of the cohomology of induced vector bundles on G/B in characteristic p

Abstract We continue the development of the homological theory of quantum general linear groups previously considered by the first author. The development is used to transfer information to the representation theory of quantised Schur algebras. The acyclicity of induction from some rank-one modules for quantised Borel–Schur subalgebras is deduced. This is used to prove the exactness of the complexes recently constructed by Boltje and Maisch, giving resolutions of the co-Specht modules for Hecke algebras.