Explicit Matrices for Irreducible Representations of Weyl Groups

We present algorithms for constructing explicit matrices for every irreducible representation of a Weyl group, with particular emphasis on the exceptional groups. The algorithms we present will produce representing matrices in either of two forms: real orthogonal, with matrix entries that are square roots of rationals, or rational and seminormal. In both cases, the matrices are “hereditary” in the sense that they behave well with respect to restriction along a chosen chain of parabolic subgroups.