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

Edouard Feingesicht

Abstract

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.

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

Abstract

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.
Paper Structure (10 sections, 47 theorems, 44 equations, 1 algorithm)

This paper contains 10 sections, 47 theorems, 44 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cycle sets
  4. Monomial matrices
  5. Braces
  6. Garsideness
  7. Dehornoy's class
  8. Non-degeneracy
  9. Bounding of the Dehornoy's class
  10. Sylow subgroups and decomposition

Key Result

Proposition 1

If $S$ is square-free and its permutation group $\mathcal{G}$ abelian then the conjecture holds.

Theorems & Definitions (125)

  • Conjecture
  • Proposition
  • Theorem
  • Definition 1.1: rump
  • Definition 1.2: rump
  • Example 1.3
  • Remark 1.4
  • Lemma 1.5
  • Corollary 1.6
  • Example 1.7
  • ...and 115 more