Representations from matrix varieties, and filtered RSK

Representations from matrix varieties, and filtered RSK

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 questio...

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Veröffentlicht in:arXiv.org 2024-03
Hauptverfasser: Price, Abigail, Stelzer, Ada, Alexander, Yong
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Bicrystals
Combinatorial analysis
Matrix algebra
Representations
Toruses
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description 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.
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subjects Bicrystals
Combinatorial analysis
Matrix algebra
Representations
Toruses
title Representations from matrix varieties, and filtered RSK
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