Matroid invariants from Schubert calculus

Abstract

Grassmannians are varieties parameterizing the k-dimensional subspaces of an n-dimensional vector space. To a point in a Grassmannian we associate two objects: A representable matroid storing combinatorial data about linear independence among the columns of a matrix whose rowspace is the point. The projective toric variety obtained as the closure of the orbit of the point under the action of an algebraic torus on the Grassmannian. We study the class of the torus orbit closure in the integral cohomology ring of the Grassmannian. Its coefficients in the Schubert basis are matroid invariants, which we call Schubert coefficients of a matroid. We show that the Schubert coefficients satisfy a linear relation and that they respect the dual matroid. If the matroid contains a loop or a coloop we describe how to obtain its Schubert coefficients from the Schubert coefficients of the matroid with the loop or coloop removed. Next we follow the recent work of Berget and Fink to define Schubert coefficients of non-representable matroids, that is, matroids that cannot be obtained from points in Grassmannians. We confirm the conjectured positivity of these Schubert coefficients in a special case. We use valuativity and results from the representable case to compute the Schubert coefficients of some non-representable matroids, namely the Fano, non-Pappus and Vamos matroids. In each case we confirm the positivity of the Schubert coefficients.