Construction of GL(n)modules using composite tableaux

Abstract

The program of constructing irreducible modules of the classical groups via Young tableaux is further extended to include mixed tensor (or rational) modules of the general linear groups GL( n). With μ a partition of u and ν a partition of ν, the GL(n)-module corresponding to irreducible representation , in Littlewood's notation, is constructed as a submodule of the mixed tensor space The bases for these irreducible GL(n)-modules are provided by the standard composite tableaux of King, while the reduction of an arbitrary composite tableaux to standard form is provided by the Garnir relations and a trace removal process, closely related to that introduced by Berele in the construction of Sp(n)-modules. The standardisation procedure is algorithmic and enables matrix representations of GL(n) and its Lie algebra gl(n) to be constructed explicitly over the integers. As with the construction of orthogonal group modules, a link between the trace condition on a composite tableau and Garnir relations on its associa...