A quasisymmetric function for matroids

Louis J. Billera, Ning Jia, Victor Reiner

Research output: Contribution to journalArticlepeer-review

17 Scopus citations

Abstract

A new isomorphism invariant of matroids is introduced, in the form of a quasisymmetric function. This invariant: •defines a Hopf morphism from the Hopf algebra of matroids to the quasisymmetric functions, which is surjective if one uses rational coefficients;•is a multivariate generating function for integer weight vectors that give minimum total weight to a unique base of the matroid;•is equivalent, via the Hopf antipode, to a generating function for integer weight vectors which keeps track of how many bases minimize the total weight;•behaves simply under matroid duality;•has a simple expansion in terms of P-partition enumerators;•is a valuation on decompositions of matroid base polytopes. This last property leads to an interesting application: it can sometimes be used to prove that a matroid base polytope has no decompositions into smaller matroid base polytopes. Existence of such decompositions is a subtle issue arising from the work of Lafforgue, where lack of such a decomposition implies that the matroid has only a finite number of realizations up to scalings of vectors and overall change-of-basis.

Original languageEnglish (US)
Pages (from-to)1727-1757
Number of pages31
JournalEuropean Journal of Combinatorics
Volume30
Issue number8
DOIs
StatePublished - Nov 1 2009

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