Enumerating Projective Reflection Groups

Abstract

Projective re ection groups have been recently dened 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 re ection groups of type G(r; p; n). In this paper we dene some statistics analogous to descent number and major index over the projective re ection 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 invariant algebras, are also treated.