Representations from matrix varieties, and filtered RSK
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.
Cite
@article{arxiv.2403.09938,
title = {Representations from matrix varieties, and filtered RSK},
author = {Abigail Price and Ada Stelzer and Alexander Yong},
journal= {arXiv preprint arXiv:2403.09938},
year = {2025}
}
Comments
Revision: adds an appendix detailing connections between the main results, prior work on Hilbert series of matrix Schubert varieties, and open problems on their minimal free resolutions. 38 pages