Abstract
We introduce the class of projective reflection groups which includes all complex reflection groups. We show that several aspects involving the combinatorics and the representation theory of complex reflection groups find a natural description in this wider setting. On introduit la classe des groupes de réflexions projectifs, ce qui généralise la notion de groupe engendré par des réflexions. On montre que plusieurs aspects concernant la combinatoire et la théorie des représentations des groupes de réflexions complexes trouvent une description naturelle dans ce cadre plus général.
Highlights
A complex reflection is an endomorphism of a complex vector space V which is of finite order and such that its fixed point space is of codimension 1
The relationship between the combinatorics and the representation theory of symmetric groups is fascinating from both combinatorial and algebraic points of view, and the problem of generalizing these sort of results to all reflection groups has been faced in many ways
Some attempts to extend these results to other reflection groups have been made, in particular for Weyl groups of type D, (see, e.g., [8; 9; 4]) though they are probably not completely satisfactory as in the case of wreath products
Summary
A complex reflection (or a reflection) is an endomorphism of a complex vector space V which is of finite order and such that its fixed point space is of codimension 1. Finite reflection groups are finite subgroups of GL(V ) generated by reflections They have probably been introduced by Shephard in [16] and have been characterized by means of their ring of invariants and completely classified by Chevalley [11] and Shephard-Todd [17] in the fifties, generalizing the well-known fundamental theorem of symmetric functions. In this classification there is an infinite family G(r, p, n) of irreducible reflection groups, where r, p, n are positive integers (with r ≡ 0 mod p) and 34 other exceptional groups.
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