Matroid Chern-Schwartz-MacPherson cycles and Tutte activities

TL;DR

This paper establishes a direct geometric method to compute matroid Chern-Schwartz-MacPherson cycle degrees, linking tropical intersection theory with Tutte polynomial coefficients and activities, advancing understanding of matroid invariants.

Contribution

It provides a direct calculation approach for matroid Chern-Schwartz-MacPherson cycle degrees using tropical geometry and Tutte activities, simplifying previous inductive methods.

Findings

Degree calculation via tropical intersection matches Tutte polynomial coefficients.

Weighted point count aligns with Gioan-Las Vergnas activities expansion.

Supports log-concavity results for Tutte polynomial coefficients.

Abstract

Lop\'ez de Medrano-Rin\'con-Shaw defined Chern-Schwartz-MacPherson cycles for an arbitrary matroid $M$ and proved by an inductive geometric argument that the unsigned degrees of these cycles agree with the coefficients of $T(M;x,0)$, where $T(M;x,y)$ is the Tutte polynomial associated to $M$. Ardila-Denham-Huh recently utilized this interpretation of these coefficients in order to demonstrate their log-concavity. In this note we provide a direct calculation of the degree of a matroid Chern-Schwartz-MacPherson cycle by taking its stable intersection with a generic tropical linear space of the appropriate codimension and showing that the weighted point count agrees with the Gioan-Las Vergnas refined activities expansion of the Tutte polynomial.