Algebra-Geometry-Combinatorics Seminar

Speaker: Ada Stelzer (UIUC)

Title: Grothendieck polynomials, determinantal varieties, and equivariant Hilbert series

Abstract: When a reductive group G acts on an embedded projective variety X, the associated coordinate ring C[X] is a G-representation. The data of this representation may be recorded directly as the G-equivariant Hilbert series of C[X], or more compactly as its K-polynomial or twisted K-polynomial (which are connected to the minimal free resolution and multidegree of C[X] respectively). Non-cancellative combinatorial rules for the coefficients in all three polynomials are therefore desirable. In this talk we focus on determinantal varieties, where the combinatorics of pipe dreams and the Robinson-Schensted-Knuth correspondence naturally arise. We present joint work with Abigail Price and Alexander Yong describing the G-equivariant Hilbert series of generalized determinantal varieties, along with open problems and directions for future research.