Representations from matrix varieties, and filtered RSK
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866908577555283968 |
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| author | Price, Abigail Stelzer, Ada Yong, Alexander |
| author_facet | Price, Abigail Stelzer, Ada Yong, Alexander |
| contents | Matrix Schubert varieties (Fulton '92) carry natural actions of Levi groups. Their coordinate rings are thereby Levi-representations; what is a combinatorial counting rule for the multiplicities of their irreducibles? When the Levi group is a torus, (Knutson-Miller '04) answers the question. We present a general solution, a common refinement of the multigraded Hilbert series, the Cauchy identity, and the Littlewood-Richardson rule. Our result applies to any ``bicrystalline'' algebraic variety; we define these using the operators of (Kashiwara '95) and of (Danilov-Koshevoi '05, van Leeuwen '06). The proof introduces a ``filtered'' generalization of the Robinson-Schensted-Knuth correspondence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_09938 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Representations from matrix varieties, and filtered RSK Price, Abigail Stelzer, Ada Yong, Alexander Representation Theory Commutative Algebra Combinatorics Matrix Schubert varieties (Fulton '92) carry natural actions of Levi groups. Their coordinate rings are thereby Levi-representations; what is a combinatorial counting rule for the multiplicities of their irreducibles? When the Levi group is a torus, (Knutson-Miller '04) answers the question. We present a general solution, a common refinement of the multigraded Hilbert series, the Cauchy identity, and the Littlewood-Richardson rule. Our result applies to any ``bicrystalline'' algebraic variety; we define these using the operators of (Kashiwara '95) and of (Danilov-Koshevoi '05, van Leeuwen '06). The proof introduces a ``filtered'' generalization of the Robinson-Schensted-Knuth correspondence. |
| title | Representations from matrix varieties, and filtered RSK |
| topic | Representation Theory Commutative Algebra Combinatorics |
| url | https://arxiv.org/abs/2403.09938 |