Representations from matrix varieties, and filtered RSK

Abigail Price; Ada Stelzer; Alexander Yong

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arXiv · 2403.09938

Representations from matrix varieties, and filtered RSK

Abigail Price·Ada Stelzer·Alexander Yong

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Abstract

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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Abigail Price, Ada Stelzer, Alexander Yong. 2024-03-15. Representations from matrix varieties, and filtered RSK. https://arxiv.org/abs/2403.09938

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