Constructing Irreducible Representations of Weyl Groups

Lee Hawkins

Séminaire lotharingien de combinatoire, Tome 34 (1995)

Résumé

We describe a construction of irreducible representations of Weyl groups based on a remarkably simple procedure given by I. G. Macdonald. For a given Weyl group W with root system R, each subsystem S of R gives rise to an irreducible representation of W. In general, however, not all the irreducible representations can be realised in this way. We show that other special subsets of R lead to representations unobtainable via the subsystem approach. The focus of this work is to determine explicitly Macdonald's representations and various computational techniques are given for finding generating sets and bases for the irreducible W-modules produced by the construction. To illustrate the success of these techniques, we enumerate examples in the Weyl group of type E₆.