Enumerating projective reflection groups
نویسندگان
چکیده
Projective reflection groups have been recently defined by the second author. They include a special class of groups denoted G(r, p, s, n) which contains all classical Weyl groups and more generally all the complex reflection groups of type G(r, p, n). In this paper we define some statistics analogous to descent number and major index over the projective reflection groups G(r, p, s, n), and we compute several generating functions concerning these parameters. Some aspects of the representation theory of G(r, p, s, n), as distribution of one-dimensional characters and computation of Hilbert series of some invariant algebras, are also treated. Résumé. Les groupes de réflexions projectifs ont été récemment définis par le deuxième auteur. Ils comprennent une classe spéciale de groupes notée G(r, p, s, n), qui contient tous les groupes de Weyl classiques et plus généralement tous les groupes de réflexions complexes du type G(r, p, n). Dans ce papier on définit des statistiques analogues au nombre de descentes et à l’indice majeur pour les groupes G(r, p, s, n), et on calcule plusieurs fonctions génératrices. Certains aspects de la théorie des représentations de G(r, p, s, n), comme la distribution des caractères linéaires et le calcul de la série de Hilbert de quelques algèbres d’invariants, sont aussi abordés.
منابع مشابه
ar X iv : 0 90 2 . 06 84 v 1 [ m at h . C O ] 4 F eb 2 00 9 PROJECTIVE REFLECTION GROUPS
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 all non exceptional irreducible complex reflection groups find a natural description in this wider setting.
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