Equivariant classes of matrix matroid varieties

TL;DR

This paper investigates the equivariant cohomology classes of matrix matroid varieties, linking their coefficients to enumerative geometry problems that generalize linear Gromov-Witten invariants.

Contribution

It introduces a method to compute the equivariant classes of matrix matroid varieties and explores their fundamental properties, connecting algebraic geometry and combinatorics.

Findings

Coefficients solve enumerative geometry problems

Provides formulas for calculating classes

Establishes properties of these classes

Abstract

Consider an integer associated with every subset of the set of columns of an $n\times k$ matrix. The collection of those matrices for which the rank of a union of columns is the predescribed integer for every subset, will be denoted by $X_C$. We study the equivariant cohomology class represented by the Zariski closure $Y_C$ of this set. We show that the coefficients of this class are solutions to problems in enumerative geometry, which are natural generalization of the linear Gromov-Witten invariants of projective spaces. We also show how to calculate these classes and present their basic properties.