A quasisymmetric function for matroids

A new isomorphism invariant of matroids is introduced, in the form of a quasisymmetric function. This invariant (1) defines a Hopf morphism from the Hopf algebra of matroids to the quasisymmetric functions, which is surjective if one uses rational coefficients, (2) is a multivariate generating function for integer weight vectors that give minimum total weight to a unique base of the matroid, (3) 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, (4) behaves simply under matroid duality, (5) has a simple expansion in terms of P-partition enumerators, and (6) 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 in work of Lafforgue, where lack of such a decomposition implies the matroid has only a finite number of realizations up to projective equivalence.

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