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

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 that has prime-power size.

Moreover, we show that, in the particular case of latin solutions, the situation is more rigid.