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Séminaire Lotharingien de Combinatoire, B34d (1995), 19pp.

# Lee Hawkins

# Constructing Irreducible Representations of Weyl Groups

**Abstract.**
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 *E*_{6}.

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