Equivariant K-Theory Classes of Matrix Orbit Closures. (2nd June 2021)

• Record Type: Journal Article
• Authors: Berget, Andrew; Fink, Alex
• Is Part Of: International mathematics research notices. Volume 2022:Number 18(2022)
• Page Start: 14105
• Page End: 14133
• Publication Date: 2021-06-02
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnab135
• Languages: English
• ISSNs: 1073-7928

Abstract

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.