Representations from matrix varieties, and filtered RSK
Abigail Price, Ada Stelzer, Alexander Yong
TL;DR
The paper tackles the problem of decomposing coordinate rings ${\mathbb C}[{\mathfrak X}]$ of ${\mathfrak X}\subseteq {\sf Mat}_{m,n}$ under reductive Levi actions, aiming to determine multiplicities of irreducible factors. It introduces a bicrystal framework via Levi subgroups ${\bf L}_{{\mathbf I}|{\mathbf J}}$ and a filtered Robinson-Schensted-Knuth correspondence ${\sf filterRSK}$, providing a combinatorial handle on multigraded Hilbert series and representation-theoretic decompositions. A generalized Cauchy identity is established: ${\chi}_{\mathbb{C}[{\mathfrak X}]} = \sum_{\underline{\lambda},\underline{\mu}} c^{\mathfrak X}_{\underline{\lambda}|\underline{\mu}} \, s_{\underline{\lambda}}(\mathbf{x}) s_{\underline{\mu}}(\mathbf{y})$, with coefficients counting highest-weight basis elements via ${\sf filterRSK}$. The results apply to unions of matrix Schubert varieties (the ${\mathfrak X}_w$) and yield a polytopal interpretation of coefficients, generalizing the Cauchy and Littlewood–Richardson rules while linking to Knutson–Miller Hilbert series and equivariant decompositions; the framework also points toward equivariant minimal free resolutions for these varieties. Overall, the work provides a unifying combinatorial and geometric toolkit for understanding ${\bf L}$-equivariant structures in matrix-variety coordinate rings and their broader algebraic-geometric consequences.
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.