Representations of Coxeter groups over fusion rings and hyperplane complements

TL;DR

This paper explores how Coxeter groups can be represented over fusion rings and their associated Vinberg systems, leading to geometric embeddings of hyperplane complements that relate to Artin--Tits groups.

Contribution

It introduces new faithful representations of Coxeter groups over fusion rings and connects them to geometric embeddings of hyperplane complements, revealing novel structures in Artin--Tits groups.

Findings

Embeddings of hyperplane complements induced by Coxeter group representations

Geometric realizations of strong admissible homomorphisms between Artin--Tits groups

New connections between Coxeter groups, fusion rings, and hyperplane arrangements

Abstract

We study faithful realisations of Coxeter groups over fusion rings and study Vinberg systems associated to them. We show that they induce embeddings of hyperplane complements, which provide geometrical realisations of certain types of strong admissible (LCM) homomorphisms between Artin--Tits groups.