Richard P. Stanley Seminar in Combinatorics: Gröbner crystal structures and equivariant Hilbert series

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

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