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 Ram\'irez and Vendramin, ask about the existence of certain indecomposable involutive solutions whose squaring maps are transpositions or 3-cycles. In this paper, we explore this problem in a more general setting by examining the case where $T$ is a $p$-cycle, for an arbitrary prime number $p$. Our results provide negative answers to the aforementioned questions under the assumption that the solution is latin or that its size is a prime-power. As a further application, we also present some decomposability theorems for solutions whose permutation groups are nilpotent.