The plactic monoid \({\textbf{P}}\) of Lascoux and Schützenberger (in: de Luca, A. (ed.), Noncommutative Structures in Algebra and Geometric Combinatorics, Proceedings of the colloquium held at Arco Felice, Naples, July 24–26, 1978. Quaderni della Ricerca Scientifica, vol. 109, pp. 129-156. Consiglio Nazionale delle Ricerche, Roma, 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, \({\textbf{P}}\) 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 \({\textbf{P}}\) via a universal property. Serrano’s (Math Z 266(2):363–392, 2010. https://doi.org/10.1007/s00209-009-0573-0) shifted plactic monoid \(\textbf{S}\) is an analogue of \({\textbf{P}}\) that governs instead the projective representation theory of \(\mathfrak {S}_n\) and the cohomology of isotropic Grassmannians. We provide a universal property for \(\textbf{S}\) , analogous to the Lascoux–Schützenberger characterization of \({\textbf{P}}\) .
