Equivariant classes of matrix matroid varieties

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.