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 language||English (US)|
|Number of pages||31|
|Journal||European Journal of Combinatorics|
|State||Published - Nov 2009|
Bibliographical noteFunding Information:
The first author was supported by the NSF grant DMS-0100323. The third author was supported by the NSF grant DMS-0245379. This work was begun while the first author was an Ordway Visitor at the University of Minnesota School of Mathematics. The authors thank Marcelo Aguiar, Saul Blanco, Harm Derksen, Sam Hsiao, Sean Keel, Kurt Luoto, Frank Sottile, David Speyer, and Jenia Tevelev for helpful conversations, and are grateful to Alexander Barvinok, Jim Lawrence and Peter McMullen for helpful suggestions on Proposition 7.2 .