Isomorphisms, automorphisms, and generalized involution models

TL;DR

This paper studies the structure and automorphisms of projective reflection groups G(r,p,q,n), providing conditions for the existence of generalized involution models and classifying specific cases where these models exist.

Contribution

It offers new criteria for isomorphisms and automorphisms of G(r,p,q,n), and classifies when these groups admit generalized involution models in various cases.

Findings

G(r,p,1,n) has a generalized involution model iff G(r,p,1,n) ≅ G(r,1,p,n)

Explicit automorphism descriptions for G(r,p,q,n) groups

Classification of generalized involution models for specific parameters

Abstract

We investigate the generalized involution models of the projective reflection groups $G(r,p,q,n)$. This family of groups parametrizes all quotients of the complex reflection groups $G(r,p,n)$ by scalar subgroups. Our classification is ultimately incomplete, but we provide several necessary and sufficient conditions for generalized involution models to exist in various cases. In the process we solve several intermediate problems concerning the structure of projective reflection groups. We derive a simple criterion for determining whether two groups $G(r,p,q,n)$ and $G(r,p',q',n)$ are isomorphic. We also describe explicitly the form of all automorphisms of $G(r,p,q,n)$, outside a finite list of exceptional cases. Building on prior work, this allows us to prove that $G(r,p,1,n)$ has a generalized involution model if and only if $G(r,p,1,n) \cong G(r,1,p,n)$. We also classify which groups $G(r,p,q,n)$ have generalized involution models when $n=2$, or $q$ is odd, or $n$ is odd.