Equivariant Chow classes of matrix orbit closures

TL;DR

This paper demonstrates that the equivariant Chow class of matrix orbit closures under a product group action is uniquely determined by the associated matroid, linking algebraic geometry with combinatorial matroid theory.

Contribution

It introduces a method to compute equivariant Chow classes of matrix orbit closures using matroids and factorial Schur polynomials, connecting geometric and combinatorial structures.

Findings

Chow class of an orbit closure is determined by the matroid.

Splitting map from Chow ring of matrices to Grassmannian is explicitly constructed.

Class of a subvariety maps to the class of matrices with row span in that subvariety.

Abstract

Let $G$ be the product $GL_r(C) \times (C^\times)^n$. We show that the $G$-equivariant Chow class of a $G$ orbit closure in the space of $r$-by-$n$ matrices is determined by a matroid. To do this, we split the natural surjective map from the $G$ equvariant Chow ring of the space of matrices to the torus equivariant Chow ring of the Grassmannian. The splitting takes the class of a Schubert variety to the corresponding factorial Schur polynomial, and also has the property that the class of a subvariety of the Grassmannian is mapped to the class of the closure of those matrices whose row span is in the variety.