An algorithm for Weyl module irreducibility

Marilena Barnabei ; V. Frontini ; F. Sgallari

Séminaire lotharingien de combinatoire, Tome 22 (1989)

Résumé

In the classical representation theory of general linear groups over fields of characteristic zero two classes of modules play a fundamental role, namely, Schur modules and Weyl modules relative to a given Young shape. As well known, these are irreducible modules, and, for every Young shape \lambda, the Schur module relative to \lambda is isomorphic to the Weyl module relative to the conjugate shape \lambda'. Recently, it has been recognized that the definitions of Schur and Weyl modules can be adapted in order to make sense over fields of arbitrary characteristics, giving rise to two classes of modules which are indecomposable but, in general, neither irreducible, nor isomorphic. Hence, the problem arises of deciding, for a given Young shape, in which characteristics the corresponding Weyl module is not irreducible. It has been shown that the solution of this problem is related to the rank of a matrix with integer entries, built up by considering Young tableaux of the given shape.

In the present paper we first exhibit some theoretical results, based on a new presentation of Weyl modules, which imply that a matrix of smaller size can be equivalently considered. Next, we present an algorithm which constructs such matrices and specifies in which characteristics there is no full rank.

The paper has been finally published under the same title in Rend. Sem. Mat. Univ. Politec. Torino 49 (1991), 217-232.