Overview
This work explores specific conditions for indecomposable involutive solutions to the Yang-Baxter equation, focusing on cases where the associated squaring map, denoted as $T$, is a $p$-cycle for an arbitrary prime number $p$. The study builds upon prior research, particularly Rump's findings concerning Gateva-Ivanova's conjecture on square-free solution decomposability, which highlighted the significance of the squaring map $T$ in understanding decomposability theorems.
Research Context
The decomposability of solutions to the Yang-Baxter equation has been a central topic in algebraic research. Rump's proof of Gateva-Ivanova's conjecture regarding square-free solutions underscored the role of the squaring map $T$. This stimulated further investigation into decomposability theorems derived from the structure of $T$. Ramírez and Vendramin posed two specific questions concerning the existence of certain indecomposable involutive solutions. These questions specifically inquired about scenarios where the squaring maps are transpositions or $3$-cycles.
Approach
The research examines the problem by considering the general case where the squaring map $T$ is a $p$-cycle, with $p$ being an arbitrary prime number. The investigation was conducted under two primary assumptions: first, that the solution has a nilpotent permutation group, and second, that the solution possesses a prime-power size.
Findings
- The study provides negative answers to the questions posed by Ramírez and Vendramin concerning the existence of indecomposable involutive solutions. These negative answers are contingent on the assumption that the solution's permutation group is nilpotent.
- Negative answers were also obtained under the assumption that the solution has a prime-power size.
- In the specific context of latin solutions, the research indicates that the situation exhibits greater rigidity.