Enumerating Projective Reflection Groups

TL;DR

This paper introduces new statistics for projective reflection groups, computes related generating functions, and explores aspects of their representation theory, including character distributions and invariant algebra Hilbert series.

Contribution

It defines new combinatorial statistics for projective reflection groups and analyzes their generating functions and representation-theoretic properties.

Findings

Derived formulas for generating functions of descent and major index statistics.

Analyzed distribution of one-dimensional characters in G(r; p; s; n).

Computed Hilbert series of invariant algebras associated with these 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.