Modular symmetric functions and irreducible modular representations of general linear groups

Abstract

The usual way to get information on the irreducible modular, defining characteristic, representations of a simple reductive algebraic group in prime characteristic is to study how the Weyl modules decompose. In this paper we explore the alternate, but classical, approach of decomposing tensor products, in the case of the general linear groups. This study singles out a particular class of tensor products, conjectured to be sufficient to determine the irreducibles. This class of tensor products may be characterized as the dominant weight spaces in the truncated coordinate ring of matrices. We are led to introduce certain modular analogues of classical symmetric functions, and to a result generalizing the fundamental theorem of symmetric functions.