We study involutive non-degenerate set-theoretic solutions $(X,r)$ of the Yang-Baxter equation on a finite set X. The emphasis is on the case where $(X,r)$ is indecomposable, so the associated permutation group $\mathcal{G}(X,r)$ acts transitively on X. One of the major problems is to determine how such solutions are built from the imprimitivity blocks; and also how to characterize these blocks. We focus on the case of so called simple solutions, which are of key importance. Several infinite families of such solutions are constructed for the first time. In particular, a broad class of simple solutions of order $p^2$, for any prime p, is completely characterized.

中文翻译：

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构造 Yang-Baxter 方程的有限简单解

我们研究对合非退化集合论解 $(十,r)$有限集X上的 Yang-Baxter 方程。重点是在这种情况下$(十,r)$ 是不可分解的，所以关联的置换群 $\mathcal{格}(十,r)$作用于X。主要问题之一是确定如何从非原始性块构建此类解决方案；以及如何表征这些块。我们专注于所谓的简单解决方案的案例，这非常重要。首次构建了多个此类解决方案的无限族。特别是，一大类简单的阶解${磷}^2$，对于任何素数p，是完全表征的。