Explicit matrices for the irreducible representations of Weyl groups

Motivated by the problem of classifying the irreducible unitary representations of real and p-adic Lie groups, we needed to compute the signature of certain operators in the group algebras of Weyl groups. To make this even remotely feasible for (say) E₈, we needed explicit matrices representing the generators in each irreducible representation. In the classical cases, it is well known how to do this, and goes back to the work of Alfred Young. However, as far as we know, this has never been done for the exceptional groups until now (at least for E₈). The solutions turn out to be surprisingly compact, and involve interesting computational and combinatorial problems.