Equivariant K-theory classes of matrix orbit closures

Abstract

The group $G = \GL_r(k) \times (k^\times)^n$ acts on $\AA^{r \times n}$, the space of $r$-by-$n$ matrices: $\GL_r(k)$ 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 $\AA^{r \times n}$ 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.