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| Format: | Preprint |
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2024
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| Online-Zugang: | https://arxiv.org/abs/2411.17619 |
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| _version_ | 1866916496758800384 |
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| author | Estupiñán-Salamanca, Santiago Pechenik, Oliver |
| author_facet | Estupiñán-Salamanca, Santiago Pechenik, Oliver |
| 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 $\mathbf{P}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17619 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A universal characterization of the shifted plactic monoid Estupiñán-Salamanca, Santiago Pechenik, Oliver Combinatorics Group Theory 05E05, 18A30, 20M05, 20M25 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 $\mathbf{P}$. |
| title | A universal characterization of the shifted plactic monoid |
| topic | Combinatorics Group Theory 05E05, 18A30, 20M05, 20M25 |
| url | https://arxiv.org/abs/2411.17619 |