A universal characterization of the shifted plactic monoid

The plactic monoid $\mathbf{P}$ of Lascoux and Schützenberger (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 Schützenberger 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-Schützenberger characterization of $\mathbf{P}$.