Abstract
Many important invariants for matroids and polymatroids, such as the Tutte polynomial, the Billera-Jia-Reiner quasi-symmetric function, and the invariant $\mathcal{G}$ introduced by the first author, are valuative. In this paper we construct the $\mathbb{Z}$-modules of all $\mathbb{Z}$-valued valuative functions for labelled matroids and polymatroids on a fixed ground set, and their unlabelled counterparts, the $\mathbb{Z}$-modules of valuative invariants. We give explicit bases for these modules and for their dual modules generated by indicator functions of polytopes, and explicit formulas for their ranks. Our results confirm a conjecture of the first author that $\mathcal{G}$ is universal for valuative invariants. Beaucoup des invariants importants des matroïdes et polymatroïdes, tels que le polynôme de Tutte, la fonction quasi-symmetrique de Billera-Jia-Reiner, et l'invariant $\mathcal{G}$ introduit par le premier auteur, sont valuatifs. Dans cet article nous construisons les $\mathbb{Z}$-modules de fonctions valuatives aux valeurs entières des matroïdes et polymatroïdes étiquetés définis sur un ensemble fixe, et leurs équivalents pas étiquetés, les $\mathbb{Z}$-modules des invariants valuatifs. Nous fournissons des bases des ces modules et leurs modules duels, engendrés par fonctions caractéristiques des polytopes, et des formules explicites donnant leurs rangs. Nos résultats confirment une conjecture du premier auteur, que $\mathcal{G}$ soit universel pour les invariants valuatifs.
Highlights
Matroids were introduced by Whitney in 1935 as a combinatorial abstraction of linear dependence of vectors in a vector space
We will stick to the definition in terms of rank functions: Definition 1.1 Suppose that X is a finite set and rk : 2X → N = {0, 1, 2, . . . }, where 2X is the set of subsets of X
Matroid polytope decompositions appeared in the work of Lafforgue ([14, 15]) on compactifications of a fine Schubert cell in the Grassmannian associated to a matroid
Summary
Matroids were introduced by Whitney in 1935 (see [22]) as a combinatorial abstraction of linear dependence of vectors in a vector space. There are many distinct but equivalent definitions of matroids and polymatroids, for example in terms of bases, independent sets, flats, polytopes or rank functions. Let S(P)M(d, r) be the set of all (poly)matroids with ground set d of rank r, A function f on S(P)M(d, r) is a (poly)matroid invariant if f (d, rk) = f (d, rk ) whenever (d, rk) and (d, rk ) are isomorphic. Matroid polytope decompositions appeared in the work of Lafforgue ([14, 15]) on compactifications of a fine Schubert cell in the Grassmannian associated to a matroid It follows from Definition 1.2 that the dual P(P)M(d, r)∨ = HomZ(P(P)M(d, r), Z) is the space of all.
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