Schubert matroids, Delannoy paths, and Speyer's invariant

Abstract

We provide a combinatorial way of computing Speyer's g-polynomial on arbitrary Schubert matroids via the enumeration of certain Delannoy paths. We define a new statistic of a basis in a matroid, and express the g-polynomial of a Schubert matroid in terms of it and internal and external activities. Some surprising positivity properties of the g-polynomial of Schubert matroids are deduced from our expression. Finally, we combine our formulas with a fundamental result of Derksen and Fink to provide an algorithm for computing the g-polynomial of an arbitrary matroid.