Séminaire Lotharingien de Combinatoire, 93B.15 (2025), 12 pp.
Abigail Price, Ada Stelzer and Alexander Yong
Filtered RSK and Matrix Schubert Varieties
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. The proof introduces a "filtered" generalization of the Robinson-Schensted-Knuth correspondence and uses the operators of
[Kashiwara '95] and of
[Danilov-Koshevoi '05, van Leeuwen '06].
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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