Indecomposable solutions of the Yang-Baxter equation with permutation group of sizes $pq$ and $p^2q$

TL;DR

This paper classifies indecomposable solutions to the Yang-Baxter equation for specific permutation groups of sizes pq and p^2q, advancing understanding of algebraic structures related to the equation.

Contribution

It provides a comprehensive classification of indecomposable solutions for certain permutation groups using recent brace classification advances.

Findings

Classified all indecomposable solutions for groups of size pq.

Classified all solutions for abelian groups of size p^2q.

Classified all solutions for dihedral groups of size p^2q.

Abstract

In this paper we study the problem of classification of indecomposable solutions of the Yang-Baxter equation. Using a scheme proposed by Bachiller, Ced\'o, and Jespers, and recent advances in the classification of braces we classify all indecomposable solutions with some particular permutation groups. We do this for all groups of size $pq$, all abelian groups of size $p^2q$ and all dihedral groups of size $p^2q$.