Models of representations of Weyl groups

A model of a representation of a group W, with simple reflections s₁,…,s_n, is a set of matrices A₁,…,A_n so that s_i → A_i generates the representation.

It is well known that every representation of a Weyl group has an integral model, i.e. each matrix A_i has integral entries. This generalizes the result for the symmetric group.

This directory contains models of all irreducible representations of Weyl groups of low rank, and software for creating and manipulating these models.

There are two types of models:

• Integral models. The matrices are integral, and the invariant form is not necessarily diagonal. These were made by Jeffrey Adams, using Magma.
• Rational seminormal models. The matrices are rational, and the invariant form is diagonal. These are by John Stembridge, using Maple. Much more information about these models may be found at Stembridge’s site Hereditary Matrix Models for Weyl Groups.

The Perl program convert.pl converts between different kinds of models. Download it and run it yourself; give the command without any arguments for a help file.