On Dehornoy's representation for the Yang-Baxter equation

2024/09/16 by Carsten Dietzel, Dietzel, Carsten, Edouard Feingesicht +3
Computer Science · Mathematics · #16T25 #20C15 #20N02 #81R50 #Advanced Algebra and Geometry #FOS: Mathematics #Graph theory and applications #Group Theory (math.GR) #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2409.10648

openalex publication_date 2024/09/16 · openalex created_date 2024/10/24 · openalex updated_date 2026/07/28

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.

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