Valuative invariants for polymatroids

Details Valuative invariants for polymatroids Valuative invariants for polymatroids

2009-08-20

arxiv.org/abs/0908.2988v2

Advances in Mathematics 225 (2010) pp. 1840-1892

Mathematics

Combinatorics

54 pp, 9 figs. Mostly minor changes; Cor 10.5 and formula for products of $u$s corrected; Prop 7.2 is new. To appear in Advanc

Scientific paper

10.1016/j.aim.2010.04.016

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 $\Z$-modules of all $\Z$-valued valuative functions for labeled matroids and polymatroids on a fixed ground set, and their unlabeled counterparts, the $\Z$-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.

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