A Combinatorial Generalisation of Rank two Complex Reflection Groups via Generators and Relations

TL;DR

This paper introduces J-reflection groups as a generalization of rank two complex reflection groups and toric reflection groups, providing uniform presentations, analyzing their centers, and classifying them up to isomorphism.

Contribution

It offers a unified presentation framework for J-reflection groups, extending known results for complex and toric reflection groups, and classifies these groups up to reflection isomorphisms.

Findings

Uniform generators and relations for J-reflection groups

Center of J-reflection groups is cyclic

Classification of J-reflection groups up to isomorphism

Abstract

Complex reflection groups of rank two are precisely the finite groups in the family of groups that we call J-reflection groups. These groups are particular cases of J-groups as defined by Achar & Aubert in 2008. The family of J-reflection groups generalises both complex reflection groups of rank two and toric reflection groups, a family of groups defined and studied by Gobet. We give uniform presentations by generators and relations of J-reflection groups, which coincide with the presentations given by Brou\'e, Malle and Rouquier when the groups are finite. In particular, these presentations provide uniform presentations for complex reflection groups of rank two where the generators are reflections (however the proof uses the classification of irreducible complex reflection groups). Moreover, we show that the center of J-reflection groups is cyclic, generalising what happens for irreducible complex reflection groups of rank two and toric reflection groups. Finally, we classify J-reflection groups up to reflection isomorphisms.