Constructing finite simple solutions of the Yang-Baxter equation

TL;DR

This paper investigates the structure of finite simple solutions to the Yang-Baxter equation, especially focusing on indecomposable cases, and constructs several infinite families including solutions of order p^2 for primes p.

Contribution

It introduces new methods to construct and characterize simple solutions of the Yang-Baxter equation, including a complete classification of solutions of order p^2.

Findings

Constructed several infinite families of simple solutions.

Characterized solutions of order p^2 for any prime p.

Provided insights into the structure of indecomposable solutions.

Abstract

We study involutive non-degenerate set-theoretic solutions (X,r) of the Yang-Baxter equation on a finite set X. The emphasis is on the case where (X,r) is indecomposable, so the associated permutation group acts transitively on X. One of the major problems is to determine how such solutions are built from the imprimitivity blocks; and also how to characterize these blocks. We focus on the case of so called simple solutions, which are of key importance. Several infinite families of such solutions are constructed for the first time. In particular, a broad class of simple solutions of order p^2, for any prime p, is completely characterized.