Plactic monoids: A braided approach

Abstract

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