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 $\mathbf{ P}$.
Categories: math.CO math.GR