Matroid Chern-Schwartz-MacPherson cycles and Tutte activities. (English) Zbl 1512.05069
Summary: L. López de Medrano et al. [Proc. Lond. Math. Soc. (3) 120, No. 1, 1–27 (2020; Zbl 1454.14013)] defined Chern-Schwartz-MacPher-son 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 \(M\). F. Ardila et al. [J. Am. Math. Soc. 36, No. 3, 727–794 (2023; Zbl 1512.05068)] 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 [E. Gioan and M. Las Vergnas, “The active bijection 2.a-decomposition of activities for matroid bases, and Tutte polynomial of a matroid in terms of beta invariants of minors”, Preprint. arXiv:1807.06516].
MSC:
| 05B35 | Combinatorial aspects of matroids and geometric lattices |
| 05C31 | Graph polynomials |
| 52B40 | Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.) |
| 14C17 | Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry |
| 14T15 | Combinatorial aspects of tropical varieties |
Keywords:
matroids; tropical intersection theory; Tutte polynomial; Tutte activities; matroid Chern-Schwartz-MacPherson cyclesReferences:
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