arXiv subjects

Edouard Feingesicht

Publications and source records attributed to Edouard Feingesicht.

4 recordsLinked to original sources

arXiv2025

How structure groups and monoids grow

The structure groups and monoids of set-theoretic solutions to the Yang-Baxter Equation can be regarded as deformations of free abelian groups resp. monoids. In this work, we obtain explicit formulae for the growth series of the structure groups and monoids of transposition and dihedral quandles, and of the structure groups of permutation quandles. These quandles provide important families of YBE solutions. The intricate nature of our formulae confirms that, while preserving many nice properties of free abelian groups, even the simplest structure groups and monoids are remarkably rich objects. We also establish some structural properties and easily computable normal forms for the monoids considered.

math.GR↗

arXiv2024

Hecke algebras for set-theoretical solutions to the Yang--Baxter equation

We define a concept of Hecke algebra for structure groups of set-theoretical solutions to the Yang--Baxter equation. As a comparison to Artin--Tits groups of spherical type, we study some properties of this construction, while also highlighting some differences that appear, which shows a difference between finite Coxeter groups and the "Coxeter-like" group introduced by Dehornoy. We also relate this definition to known constructions on solutions (retractions). Finally, we study a particular case related to Torus Knot groups and Complex Reflexion groups.

math.QA↗

arXiv2024

On Dehornoy's representation for the Yang-Baxter equation

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.

math.GR↗

arXiv2024

Dehornoy's class and Sylows for set-theoretical solutions of the Yang-Baxter equation

We explain how the germ of the structure group of a cycle set decomposes as a product of its Sylow-subgroups, and how this process can be reversed to construct cycle sets from ones with coprime classes. We study Dehornoy's class associated to a cycle set, and conjecture a bound that we prove in a specific case. We combine the use of braces and a monomial representation, in particular to answer a question by Dehornoy on retrieving the Garside structure without a theorem of Rump, while also retrieving said theorem.

math.GR↗