Major Indices and Perfect Bases for Complex Reflection Groups

TL;DR

This paper demonstrates that complex reflection groups can be decomposed into cyclic subgroups and extends the major index and Hilbert series concepts to these groups, broadening their algebraic understanding.

Contribution

It introduces a decomposition of complex reflection groups into cyclic subgroups and extends major index and Hilbert series identities to these groups.

Findings

Decomposition of $G(r,p,n)$ into cyclic subgroups under mild conditions

Extension of major index to complex reflection groups

Hilbert series identities for these groups

Abstract

It is shown that, under mild conditions, a complex reflection group $G(r,p,n)$ may be decomposed into a set-wise direct product of cyclic subgroups. This property is then used to extend the notion of major index and a corresponding Hilbert series identity to these and other closely related groups.