Normal Reflection Subgroups of Complex Reflection Groups

TL;DR

This paper investigates normal reflection subgroups of 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 prior results by Orlik, Solomon, and the second author.

Findings

Refined theorem for fixed-space dimension generating function

Uniform proof applicable to a broader class of groups

Generalization of recent results by the second author

Abstract

We study normal reflection subgroups of complex reflection groups. Our approach 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.