Finite resolutions of modules for reductive algebraic groups

Abstract

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.