Artin groups and Yokonuma-Hecke algebras

TL;DR

This paper introduces a new algebraic construction linking Coxeter systems, Artin groups, and Yokonuma-Hecke algebras, providing insights into their structure, representations, and generalizations to complex reflection groups.

Contribution

It constructs a new algebra extension for Coxeter systems, establishes morphisms from Artin groups, and extends the framework to complex reflection groups.

Findings

The algebra C_W is a free module of finite rank for finite W.

When W is a Weyl group, C_W maps to Yokonuma-Hecke algebra.

The construction generalizes to complex reflection groups.

Abstract

We attach to every Coxeter system (W,S) an extension C_W of the corresponding Iwahori-Hecke algebra. We construct a 1-parameter family of (generically surjective) morphisms from the group algebra of the corresponding Artin group onto C_W. When W is finite, we prove that this algebra is a free module of finite rank which is generically semisimple. When W is the Weyl group of a Chevalley group, C_W naturally maps to the associated Yokonuma-Hecke algebra. When W = S_n this algebra can be identified with a diagram algebra called the algebra of `braids and ties'. The image of the usual braid group in this case is investigated. Finally, we generalize our construction to finite complex reflection groups, thus extending the Broue-Malle-Rouquier construction of a generalized Hecke algebra attached to these groups.