Plactic monoids: a braided approach

TL;DR

This paper demonstrates that the plactic monoid's product, derived from Young tableaux, can be fully characterized by a braiding on columns, linking combinatorics with solutions to the Yang--Baxter equation and cohomology.

Contribution

It introduces a braiding-based framework to understand the plactic monoid and connects its Hochschild cohomology to braided cohomology, providing new algebraic insights.

Findings

The tableaux product is determined by a braiding on columns.

Hochschild cohomology of the plactic monoid is identified with braided cohomology.

The braiding commutes with crystal reflection operators.

Abstract

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~$s\_i$.