Equivariant K-Theory Classes of Matrix Orbit Closures. (2nd June 2021) - Berget, Andrew
Abstract: The group $G = \textrm{GL}_r(k) \times (k^\times )^n$ acts on $\textbf{A}^{r \times n}$, the space of $r$ -by-$n$ matrices: $\textrm{GL}_r(k)$ acts by row operations and $(k^\times )^n$ scales columns. A matrix orbit closure is the Zariski closure of a point orbit for this action. We prove that the class of such an orbit closure in $G$ -equivariant $K$ -theory of $\textbf{A}^{r \times n}$ is determined by the matroid of a generic point. We present two formulas for this class. The key to the proof is to show that matrix orbit closures have rational singularities.