Reflection groups acting on their hyperplanes

TL;DR

This paper explores the structure of reflection groups acting on hyperplanes, revealing their relationships with braid group representations and quadratic forms, and establishing bounds on the number of hyperplanes for irreducible cases.

Contribution

It introduces a canonical connection between reflection group actions on hyperplanes and braid group representations, and proves new bounds and properties for irreducible reflection groups.

Findings

The action of reflection groups on hyperplanes relates to natural braid group representations.

Squares of defining forms span quadratic forms, implying hyperplane count bounds.

W-equivariance links to the periodicity of the representation family.

Abstract

After having established elementary results on the relationship between a finite complex (pseudo-)reflection group W < GL(V) and its reflection arrangement A, we prove that the action of W on A is canonically related with other natural representations of W, through a `periodic' family of representations of its braid group. We also prove that, when W is irreducible, then the squares of defining linear forms for A span the quadratic forms on V, which imply |A| >= n(n+1)/2 for n = dim V, and relate the W-equivariance of the corresponding map with the period of our family.