On Dehornoy's representation for the Yang-Baxter equation

TL;DR

This paper explores Dehornoy's monomial representations related to the Yang-Baxter equation, establishing their connection to the indecomposability of solutions and providing explicit constructions for indecomposable cases.

Contribution

It proves the equivalence between irreducibility of representations and indecomposability of solutions, except in a special case, and constructs explicit one-dimensional induced representations.

Findings

Irreducibility of monomial representations is equivalent to solution indecomposability, except when Dehornoy class is two.

Indecomposable solutions induce representations from explicit one-dimensional representations.

The study links algebraic properties of solutions to their associated monomial representations.

Abstract

This article investigates Dehornoy's monomial representations for structure groups and Coxeter-like groups associated with a set-theoretic solution to the Yang--Baxter equation. Using the brace structure of these groups and the language of cycle sets, we prove that the irreducibility of the associated monomial representations is equivalent to the indecomposability of the underlying solutions, except when the Dehornoy class is two. For indecomposable solutions, we show that these representations are induced from certain explicitly constructed one-dimensional representations.