A family of irretractable square-free solutions of the Yang-Baxter equation

TL;DR

This paper introduces a new family of irretractable, square-free solutions to the Yang-Baxter equation, providing counterexamples to a longstanding conjecture and exploring their algebraic properties.

Contribution

It constructs a novel family of solutions that are irretractable and square-free, including counterexamples to Gateva-Ivanova's Strong Conjecture and answering an open question.

Findings

Includes a recent counterexample to the Strong Conjecture.

Shows the structure groups are not poly-(infinite cyclic) groups.

Proves the permutation group's socle is trivial.

Abstract

A new family of non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation is constructed. All these solutions are strong twisted unions of multipermutation solutions of multipermutation level at most two. A large subfamily consists of irretractable and square-free solutions. This subfamily includes a recent example of Vendramin, who first gave a counterexample to Gateva-Ivanova's Strong Conjecture. All the solutions in this subfamily are new counterexamples to Gateva-Ivanova's Strong Conjecture and also they answer a question of Cameron and Gateva-Ivanova. It is proved that the natural left brace structure on the permutation group of the solutions in this family has trivial socle. Properties of the permutation group and of the structure group associated to these solutions are also investigated. In particular, it is proved that the structure groups of finite solutions in this subfamily are not poly-(infinite cyclic) groups.