Normal Reflection Subgroups

TL;DR

This paper investigates normal reflection subgroups within complex reflection groups, refining existing theorems to provide a more general and uniform proof involving generating functions and generalized exponents.

Contribution

It offers a refined and generalized proof of a theorem related to fixed-space dimensions in complex reflection groups, extending previous results by Orlik, Solomon, and the second author.

Findings

Refined theorem for fixed-space dimension generating functions

Uniform proof applicable to complex reflection groups

Generalization of recent results on reflection subgroups

Abstract

We study normal reflection subgroups of complex reflection groups. Our point of view leads to a refinement of a theorem of Orlik and Solomon to the effect that the generating function for fixed-space dimension over a reflection group is a product of linear factors involving generalized exponents. Our refinement gives a uniform proof and generalization of a recent theorem of the second author.