Schubert coefficients of sparse paving matroids

TL;DR

This paper investigates Schubert coefficients associated with matroids, providing formulas for disconnected and sparse paving matroids, and confirms the non-negativity conjecture for these cases.

Contribution

It introduces methods to compute Schubert coefficients for disconnected and sparse paving matroids, supporting the non-negativity conjecture.

Findings

Schubert coefficients for disconnected matroids are expressed via their connected components.

Explicit computation of Schubert coefficients for all sparse paving matroids.

Confirmation of non-negativity of Schubert coefficients in these cases.

Abstract

The Chow class of the closure of the torus orbit of a point in a Grassmannian only depends on the matroid associated to the point. The Chow class can be extended to a matroid invariant of arbitrary matroids. We call the coefficients appearing in the expansion of the Chow class in the Schubert basis the Schubert coefficients of the matroid. These Schubert coefficients are conjectured by Berget and Fink to be non-negative. We compute the Schubert coefficients of a disconnected matroid in terms of the Schubert coefficients of its connected components. And we compute the Schubert coefficients for all sparse paving matroids, and confirm their non-negativity.