2024/03/15 by Abigail Price, Price, Abigail, Ada Stelzer +3
Computer Science · #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Polynomial and algebraic computation #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2403.09938
openalex publication_date 2024/03/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/03
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.