On indecomposable involutive solutions to the Yang-Baxter equation whose squaring map is a $p$-cycle

Abstract

The pioneering work of Rump, which proved Gateva-Ivanova's conjecture concerning the decomposability of square-free solutions to the Yang-Baxter equation, significantly motivated further research into the associated squaring map $T$. This line of inquiry has yielded numerous decomposability theorems based on the underlying structure of $T$. Two seminal questions, posed by Ramírez and Vendramin, ask about the existence of certain indecomposable involutive solutions whose squaring maps are transpositions or $3$-cycles. In this paper, we explore these problems by examining the case where $T$ is a $p$-cycle, for an arbitrary prime number $p$. We provide negative answers to the aforementioned questions under the assumption that the solution has nilpotent permutation group or has prime-power cardinality, providing also some decomposition criteria in this setting. Moreover, we show that, in the particular case of latin solutions, the situation is more rigid.

10 pages + References, v4: In the first version, we found an error in the proof of the main results. The classification results now focus on the cases p=2,3 (the conclusion remains the same, but the proofs are completely different from those in the first version), and we solve partially the case p>3. Added decomposability criteria