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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.17619 |
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Table of Contents:
- 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 $\mathbf{P}$.