Workshop on Combinatorics of Enumerative Geometry

Abstract: Suppose $X$ is the affine cone of a projective variety. The Hilbert series of the coordinate ring $C[X]$ is the character of an algebraic torus. More generally, one considers a reductive algebraic group $G$ acting rationally on $X$. When $X$ is matrix space $Mat(m, n), G=GL_m \times GL_n$ acts by row and column operations. The relationship between the Hilbert series and the class of $C[Mat(m, n)]$ in the representation ring of $G$ is the Cauchy identity; its combinatorial explanation is the Robinson-Schensted-Knuth (RSK) correspondence. We study $X$ in $Mat_{m, n}$ where $G=GL_m \times GL_n$ or a Levi subgroup acts, and there is an additional compatibility of Grobner basis theory with Kashiwara’s crystal basis theory. For such ``bicrystalline’’ varieties, we give a common generalization of the Hilbert series, the Cauchy identity, and the Littlewood-Richardson rule. Our work introduces a ``filtered’’ generalization of RSK. Our main application is to determinantal varieties such as Fulton’s matrix Schubert varieties. This is joint work with Abigail Price (UIUC) and Ada Stelzer (UIUC); arXiv:2403.09938.