A quasisymmetric function for matroids

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.