Plactic monoids: A braided approach

摘要

Young tableaux carry an associative product, described by the Schenstedalgorithm. They thus form a monoid $mathbf{Pl}$, called emph{plactic}. It iscentral 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 aemph{braiding} is a set-theoretic solution to the Yang--Baxter equation. As anapplication, we identify the Hochschild cohomology of $mathbf{Pl}$, whichresists classical approaches, with the more accessible braided cohomology of$(mathbf{Col},sigma)$. The cohomological dimension of $mathbf{Pl}$ isobtained as a corollary. Also, the braiding~$sigma$ is proved to commute withthe classical crystal reflection operators~$s_i$.