Matroid Chern-Schwartz-MacPherson cycles and Tutte activities

Abstract

Lopéz de Medrano-Rincón-Shaw defined Chern-Schwartz-MacPher-son cycles for an arbitrary matroid M {M} and proved by an inductive geometric argument that the unsigned degrees of these cycles agree with the coefficients of T ( M ; x , 0 ) T({M};x,0) , where T ( M ; x , y ) T({M};x,y) is the Tutte polynomial associated to M {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.