Harm Derksen ; Alex Fink - Valuative invariants for polymatroids

dmtcs:2849 - Discrete Mathematics & Theoretical Computer Science, January 1, 2010, DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010) - https://doi.org/10.46298/dmtcs.2849
Valuative invariants for polymatroidsArticle

Authors: Harm Derksen 1; Alex Fink 2

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.


Volume: DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010)
Section: Proceedings
Published on: January 1, 2010
Imported on: January 31, 2017
Keywords: polymatroids,polymatroid polytopes,decompositions,valuations,[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO],[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]
Funding:
    Source : OpenAIRE Graph
  • Invariant Theory and Algebraic Combinatorics; Funder: National Science Foundation; Code: 0901298
  • CAREER: Invariant Theory, Algorithms and Applications; Funder: National Science Foundation; Code: 0349019

25 Documents citing this article

Consultation statistics

This page has been seen 198 times.
This article's PDF has been downloaded 335 times.