Automorphisms and Generalized Involution Models of Finite Complex Reflection Groups

TL;DR

This paper characterizes when finite complex reflection groups have generalized involution models, provides explicit automorphism formulas for certain groups, and constructs new Gelfand models for specific cases.

Contribution

It establishes necessary and sufficient conditions for the existence of generalized involution models in finite complex reflection groups and introduces new Gelfand models for these groups.

Findings

Characterization of groups with generalized involution models

Explicit automorphism formulas for G(r,p,n) groups

New Gelfand models for groups with gcd(p,n)=1

Abstract

We prove that a finite complex reflection group has a generalized involution model, as defined by Bump and Ginzburg, if and only if each of its irreducible factors is either $G(r,p,n)$ with $\gcd(p,n)=1$; $G(r,p,2)$ with $r/p$ odd; or $G_{23}$, the Coxeter group of type $H_3$. We additionally provide explicit formulas for all automorphisms of $G(r,p,n)$, and construct new Gelfand models for the groups $G(r,p,n)$ with $\gcd(p,n)=1$.