Schubert coefficients of sparse paving matroids
Jon Pål Hamre
TL;DR
The paper develops a matroidal Schubert calculus by interpreting the Chow class of torus orbit closures in Grassmannians as a valuative matroid invariant, with Schubert coefficients $d_\\lambda(M)$ encoding its expansion in the Schubert basis. It proves a structural formula for disconnected matroids and provides a complete computation of Schubert coefficients for sparse paving matroids, establishing non-negativity and revealing a canonical pattern: for connected sparse paving $M$, $d_\\lambda(M)=d_\\lambda(U_{r,n})$ for all $\lambda\neq h^c$ while $d_{h^c}(M)=\beta(M)$. A conjecture posits that sparse paving mats are precisely those with this property, supported by a detailed treatment via matroid polytopes, volumes, and subdivision arguments; the work also clarifies limitations through a Panhandle example. Together, these results connect matroid theory, Grassmannian geometry, and Schubert calculus, advancing the Berget–Fink positivity program and offering tools for computing matroid invariants in a geometric setting.
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.