Involutory reflection groups and their models

TL;DR

This paper introduces a uniform combinatorial model for involutory complex reflection groups, including Coxeter groups, linking their representation theory to involution counts.

Contribution

It constructs a comprehensive combinatorial framework for all non-exceptional involutory complex reflection groups, expanding understanding of their representations.

Findings

Established a connection between involution counts and representation sums.

Developed a uniform combinatorial model applicable to multiple group families.

Included all infinite Coxeter groups in the involutory classification.

Abstract

A finite subgroup of $GL(n,\mathbb C)$ is involutory if the sum of the dimensions of its irreducible complex representations is given by the number of absolute involutions in the group. A uniform combinatorial model is constructed for all non-exceptional irreducible complex reflection groups which are involutory including, in particular, all infinite families of finite irreducible Coxeter groups.