Young tableaux carry an associative product, described by the Schensted algorithm. They thus form a monoid $\mathbf{Pl}$, called \emph{plactic}. It is central in numerous combinatorial and algebraic applications. In this paper, the tableaux product is shown to be completely determined by a braiding $\sigma$ on the (much simpler!) set of columns $\mathbf{Col}$. Here a \emph{braiding} is a set-theoretic solution to the Yang--Baxter equation. As an application, we identify the Hochschild cohomology of $\mathbf{Pl}$, which resists classical approaches, with the more accessible braided cohomology of $(\mathbf{Col},\sigma)$. The cohomological dimension of $\mathbf{Pl}$ is obtained as a corollary. Also, the braiding~$\sigma$ is proved to commute with the classical crystal reflection operators~$s\_i$.

中文翻译：

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塑料幺半群：编织方法

年轻的画面带有一个关联产品，由 Schensted 算法描述。因此，它们形成了幺半群 $\mathbf{Pl}$，称为 \emph{plactic}。它是众多组合和代数应用的核心。在本文中，表乘积被证明完全由（简单得多！）列 $\mathbf{Col}$ 上的编织 $\sigma$ 确定。这里的 \emph {braiding} 是 Yang--Baxter 方程的集合论解。作为一个应用，我们确定了 $\mathbf{Pl}$ 的 Hochschild 上同调，它抵制了经典方法，以及更容易理解的 $(\mathbf{Col},\sigma)$ 的编织上同调。$\mathbf{Pl}$ 的上同调维数是作为推论获得的。此外，编织~$\sigma$ 被证明与经典的晶体反射算子~$s\_i$ 相通。