Valuative invariants for large classes of matroids

Abstract

We study an operation in matroid theory that allows one to transition a given matroid into another with more bases via relaxing a \emph{stressed subset}. This framework provides a new combinatorial characterization of the class of split matroids. Moreover, it permits to describe an explicit matroid subdivision of a hypersimplex, which in turn can be used to write down concrete formulas for the evaluations of any valuative invariant on these matroids. This shows that evaluations on split matroids depend solely on the behavior of the invariant on tractable subclass of Schubert matroids. We address systematically the consequences of our approach for several invariants. They include the volume and Ehrhart polynomial of base polytopes, the Tutte polynomial, Kazhdan--Lusztig polynomials, the Whitney numbers of the first and second kind, spectrum polynomials and a generalization of these by Denham, chain polynomials and Speyer's $g$-polynomials, as well as Chow rings of matroids and their Hilbert--Poincaré series. The flexibility of this setting allows us to give a unified explanation for several recent results regarding the listed invariants; furthermore, we emphasize it as a powerful computational tool to produce explicit data and concrete examples.

68 pages. To appear in J. Lond. Math. Soc

References in corpus (13)

• Matroid polytopes, nested sets and Bergman fans
• Matroids are not Ehrhart positive
• Kazhdan-Lusztig polynomials of matroids: a survey of results and conjectures
• Hilbert-Poincaré series of matroid Chow rings and intersection cohomology
• Stressed hyperplanes and Kazhdan-Lusztig gamma-positivity for matroids
• Matroid relaxations and Kazhdan-Lusztig non-degeneracy
• Ehrhart polynomials of rank two matroids
• Chow rings of matroids are Koszul
• Kazhdan-Lusztig polynomials of braid matroids
• The Merino--Welsh conjecture for split matroids
• Essence of independence: Hodge theory of matroids since June Huh
• Equivariant Kazhdan-Lusztig theory of paving matroids
• Chain enumeration, partition lattices and polynomials with only real roots

Cited by in corpus (4)

• Deletion formulas for equivariant Kazhdan-Lusztig polynomials of matroids
• Matroids in OSCAR
• Schubert matroids, Delannoy paths, and Speyer's invariant
• Face enumeration for split matroid polytopes