Equivariant K-Theory Classes of Matrix Orbit Closures

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.

Tautological classes of matroids

“…When the characteristic of is zero, so that tools like resolution of singularities and Kawamata-Viehweg vanishing are available, parts of the results here have been established in previous literature [BF22,KT09]. For instance, [BF22, Theorem C] states that any Schur functor applied to S ∨ L has vanishing higher cohomologies.…”