A universal characterization of the shifted plactic monoid

TL;DR

This paper establishes a universal property characterization of the shifted plactic monoid, extending the classical plactic monoid's intrinsic description to the shifted case, relevant for representation theory and geometry.

Contribution

It provides the first intrinsic universal property characterization of the shifted plactic monoid, paralleling the classical case.

Findings

Universal property for the shifted plactic monoid is established.

The characterization links shifted tableaux with algebraic structures.

This work connects combinatorics with representation theory and geometry.

Abstract

The plactic monoid $\mathbf{P}$ of Lascoux and Sch\"{u}tzenberger (1981) plays an important role in proofs of the Littlewood-Richardson rule for computing multiplicities in the linear representation theory of the symmetric group $\mathfrak{S}_n$ and the cohomology of Grassmannians. Commonly, $\mathbf{P}$ is defined as a quotient of a free monoid by relations derived from a careful analysis of Schensted's insertion algorithm and the jeu de taquin algorithm on semistandard Young tableaux. However, Lascoux and Sch\"{u}tzenberger also gave an intrinsic characterization of $\mathbf{P}$ via a universal property. Serrano's (2010) shifted plactic monoid $\mathbf{S}$ is an analogue of $\mathbf{P}$ that governs instead the projective representation theory of $\mathfrak{S}_n$ and the cohomology of isotropic Grassmannians. We provide a universal property for $\mathbf{S}$, analogous to the Lascoux-Sch\"{u}tzenberger characterization of $\mathbf{P}$.