A quasisymmetric function for matroids

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 PP-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.