Speaker: Ada Stelzer (UIUC)

Title: Representations from matrix varieties, and filtered RSK

Abstract: Matrix Schubert varieties [Fulton '92] are closed under the Levi group action of block-diagonal matrix multiplication, inducing a decomposition of their coordinate rings into irreducible representations. When the Levi group is a torus, [Knutson-Miller '04] gives a combinatorial rule for the multiplicity of the irreducibles in these coordinate rings. We give a combinatorial rule for the general Levi case, a common refinement of the multigraded Hilbert series, the Cauchy identity, and the Littlewood-Richardson rule. Our arguments apply to a broader class of "bicrystalline" varieties, which we define using operators of [Kashiwara '95] and [Danilov-Koshevoi '05, van Leeuwen '06]. The proof introduces a "filtered" generalization of the Robinson-Schensted-Knuth correspondence. This is joint work with Abigail Price and Alexander Yong.