Decomposition theorems for involutive solutions to the Yang-Baxter equation

TL;DR

This paper presents new decomposability theorems for involutive solutions to the Yang-Baxter equation, expanding on prior work by analyzing the cycle structure of associated permutations.

Contribution

It introduces several novel decomposability theorems based on permutation cycle structures, generalizing previous results on square-free solutions.

Findings

New decomposability theorems for involutive solutions

Cycle structure analysis as a key tool

Extension of Rump's proof to broader classes

Abstract

Motivated by the proof of Rump of a conjecture of Gateva-Ivanova on the decomposability of square-free solutions to the Yang-Baxter equation, we present several other decomposability theorems based on the cycle structure of a certain permutation associated with the solution.