2006/06/26 by Louis J. Billera, Ning Jia, Billera, Louis J. +3
Mathematics · #05B35 #52B40 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.math/0606646
openalex publication_date 2006/06/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.