A "Staircase" formula for the Chern-Schwartz-MacPherson cycle of a matroid
Franquiz Caraballo Alba, Jeffery Liu
TL;DR
This paper introduces a new 'staircase' formula for the Chern-Schwartz-MacPherson cycle of any matroid, extending known results from realizable matroids to all matroids and confirming a conjecture.
Contribution
It provides a universal formula for the CSM cycle of matroids, derived from realizable cases and proven for all, advancing understanding of matroid invariants.
Findings
The formula is valid for all matroids.
It satisfies a contraction-deletion property.
Confirms a conjecture by Fife and Rincón.
Abstract
We provide a formula for the Poincar\'e dual of the Chern-Schwartz-MacPherson (CSM) cycle of a matroid in the Chow ring of the matroid. We derive the formula from the case of matroids realizable over the complex numbers and prove that it satisfies a contraction-deletion formula. From this fact, we prove it holds for all matroids, confirming a conjecture of Fife and Rinc\'on.
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Taxonomy
TopicsAdvanced Differential Equations and Dynamical Systems · Stochastic processes and statistical mechanics · Cellular Automata and Applications