论文标题
无方数的杨巴克斯特方程的不可塑性解决方案
Indecomposable solutions of the Yang-Baxter equation of square-free cardinality
论文作者
论文摘要
研究了不可分解的涉及的非脱位set理论解决方案$（x，r）yang-baxter的基数$ p_1 \ cdots p_n $，对于不同的素数$ p_1 $ p_1，\ ldots，p_n $。事实证明，它们是$ \ leq n $级别的多电器解决方案。特别是，没有简单的无prime无正方形的基数解决方案。这解决了[F。 Cedó，J。Okniński，构建Yang-Baxter方程的有限简单解决方案，Adv。数学。 391（2021），107968]，并为溶液的不可分解性提供了一些早期的结果。这些证明基于对置换组$ \ MATHCAL G（x，r）$相关的置换组结构的详细研究。事实证明，$ p_1，\ ldots，p_n $是将$ \ Mathcal {g}（x，r）$的订单划分的唯一素数。此外，Sylow $ p_i $ -subGroups的$ \ Mathcal {g}（x，r）$是基本的Abelian $ p_i $ -groups，如果$ p_i $表示Sylow $ p_i $ p_i $ -subgroup，则是左Brace $ \ Mathcal $ \ Mathcal $ flass $ \ Mathcal $ \ g} $的$ p_i $ -sbroup。 s_n $使得$ p_ {σ（1）}，\，\，p_ {σ（1）} p_ {σ（2）}，\ dots，p_ {σ（1）} p_ {σ（2）} \ cdots p_ cdots p_ cdots p _ cdots p_ {σ（n）} $是左BRACE $ and $ nmats $ and can $ and cal c \ v \ g = $ \ MATHCAL {G}（X，R）= P_1P_2 \ CDOTS P_N $。此外，对于每个非负整数$ n $，构建了$ n $的多个级别的多孔$ p_1 \ cdots p_n $的不可分解的解决方案。
Indecomposable involutive non-degenerate set-theoretic solutions $(X,r)$ of the Yang-Baxter equation of cardinality $p_1\cdots p_n$, for different prime numbers $p_1,\ldots, p_n$, are studied. It is proved that they are multipermutation solutions of level $\leq n$. In particular, there is no simple solution of a non-prime square-free cardinality. This solves a problem stated in [F. Cedó, J. Okniński, Constructing finite simple solutions of the Yang-Baxter equation, Adv. Math. 391 (2021), 107968] and provides a far reaching extension of several earlier results on indecomposability of solutions. The proofs are based on a detailed study of the brace structure on the permutation group $\mathcal G(X,r)$ associated to such a solution. It is proved that $p_1,\ldots, p_n$ are the only primes dividing the order of $\mathcal{G}(X,r)$. Moreover, the Sylow $p_i$-subgroups of $\mathcal{G}(X,r)$ are elementary abelian $p_i$-groups and if $P_i$ denotes the Sylow $p_i$-subgroup of the additive group of the left brace $\mathcal{G}(X,r)$, then there exists a permutation $σ\in S_n$ such that $P_{σ(1)}, \, P_{σ(1)}P_{σ(2)}, \dots , P_{σ(1)}P_{σ(2)}\cdots P_{σ(n)}$ are ideals of the left brace $\mathcal{G}(X,r)$ and $\mathcal{G}(X,r)=P_1P_2\cdots P_n$. In addition, indecomposable solutions of cardinality $p_1\cdots p_n$ that are multipermutation of level $n$ are constructed, for every nonnegative integer $n$.
