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Commutative Algebra to Representation Theory, Through the Combinatorics of Filtered RSK

Presenter
February 6, 2025
Abstract
Suppose X is the affine cone of a projective variety. The Hilbert series of the coordinate ring C[X] is the character of an algebraic torus. More generally, one considers a reductive algebraic group Gacting rationally on X. When X is matrix space Mat(m,n),G=GLm×GLn acts by row and column operations. The relationship between the Hilbert series and the class of C[Mat(m,n)]in the representation ring of G is the Cauchy identity; its combinatorial explanation is the Robinson-Schensted-Knuth (RSK) correspondence. We study X in Matm,n where G=GLm×GLn or a Levi subgroup acts, and there is an additional compatibility of Grobner basis theory with Kashiwara’s crystal basis theory. For such ``bicrystalline’’ varieties, we give a common generalization of the Hilbert series, the Cauchy identity, and the Littlewood-Richardson rule. Our work introduces a ``filtered’’ generalization of RSK. Our main application is to determinantal varieties such as Fulton’s matrix Schubert varieties. This is joint work with Abigail Price (UIUC) and Ada Stelzer (UIUC); arXiv:2403.09938.