Commutative Algebra to Representation Theory Through the Combinatorics of Filtered RSK

Watch a mathematics lecture exploring the connections between commutative algebra and representation theory through combinatorial analysis. Delve into the study of affine cones of projective varieties and their Hilbert series as characters of algebraic tori, with special focus on matrix spaces and group actions. Learn about the relationship between Hilbert series and representation ring classes through the Cauchy identity and Robinson-Schensted-Knuth correspondence. Examine bicrystalline varieties where Grobner basis theory aligns with Kashiwara's crystal basis theory, leading to a unified understanding of Hilbert series, Cauchy identity, and the Littlewood-Richardson rule. Discover a new filtered generalization of RSK and its applications to determinantal varieties, including Fulton's matrix Schubert varieties, based on joint research with Abigail Price and Ada Stelzer from the University of Illinois, Urbana-Champaign.