Chow classes of matroids and standard Young tableaux

TL;DR

This paper introduces a combinatorial method to compute Chow classes of matroids, linking them to ribbon Schur functions and standard Young tableaux, with applications to permutation enumeration and matroid volumes.

Contribution

It identifies the Poincaré dual of snake matroid Chow classes with ribbon Schur functions, providing explicit formulas and extending results to all matroids via valuativity.

Findings

Explicit formula for Chow classes of snake matroids.

Connection between Schubert coefficients and standard Young tableaux.

Positivity of certain Schubert coefficients for paving matroids.

Abstract

We study the Chow classes of arbitrary matroids in the Grassmannian. We develop a new combinatorial approach for computing them, by first focusing on snake matroids and then extending our results via valuativity to any matroid. Our main contribution identifies the Poincar\'e dual of the Chow class of a snake matroid with a specific ribbon Schur function, providing an explicit formula for its coefficients in the Schubert basis as the number of standard Young tableaux of a given shape with a prescribed descent set. This agrees with a formula by Klyachko for the uniform matroid. As consequences, we recover and simplify classical results such as Gessel and Viennot's enumeration of permutations with fixed descent sets, and formulas for the volume of lattice path matroids. Furthermore, we demonstrate the power of our findings by proving that certain Schubert coefficients are positive for all connected paving matroids.