On indecomposable involutive solutions to the Yang-Baxter equation whose squaring map is a $p$-cycle
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| Format: | Preprint |
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2025
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| _version_ | 1866908477006282752 |
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| author | Castelli, Marco Kanrar, Arpan |
| author_facet | Castelli, Marco Kanrar, Arpan |
| contents | 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 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_01613 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On indecomposable involutive solutions to the Yang-Baxter equation whose squaring map is a $p$-cycle Castelli, Marco Kanrar, Arpan Quantum Algebra Group Theory 16T25, 81R50 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 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. |
| title | On indecomposable involutive solutions to the Yang-Baxter equation whose squaring map is a $p$-cycle |
| topic | Quantum Algebra Group Theory 16T25, 81R50 |
| url | https://arxiv.org/abs/2508.01613 |