Séminaire Lotharingien de Combinatoire, B22d (1989).
[Formerly: Publ. I.R.M.A. Strasbourg, 1990, 414/S-22, p.
5-16.]
Marilena Barnabei, V. Frontini and F. Sgallari
An algorithm for Weyl module irreducibility
Abstract.
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.