  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.
