On Dehornoy's representation for the Yang-Baxter equation
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.
Cite
@article{arxiv.2409.10648,
title = {On Dehornoy's representation for the Yang-Baxter equation},
author = {Carsten Dietzel and Edouard Feingesicht and Silvia Properzi},
journal= {arXiv preprint arXiv:2409.10648},
year = {2026}
}
Comments
13 pages, Comments Welcome! ; Changes in v2: new title ; Changes in v3: Introduction and abstract revised; typos corrected and notation and wording refined throughout; Proposition 2.12 added for completeness; proof of Theorem 3.2 revised; statement of Theorem 4.3 updated