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, and • 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 scalings of vectors and overall change-of-basis.

Algebraic and Geometric Methods in Enumerative Combinatorics

“…Many other natural matroid functions were later discovered to be valuative. [AFR10,BJR09,Der09]. An example of a very general valuation on matroid polytopes from [AFR10] is the formal sum R(M ) = A⊆E R A,r(A) of symbols of the form R S,k where S is a subset of E and k is an integer.…”

Ehrhart Polynomials of Matroid Polytopes and Polymatroids

“…This proves that the Ehrhart polynomial cannot be computed using deletion and contractions, as is the case for the Tutte polynomial. Examples BJR1 and BJR2 show that the Ehrhart polynomial of a matroid may help to distinguish nonisomorphic matroids: These two matroids are not isomorphic yet they have the same Tutte polynomials and the same quasi-symmetric function studied in [8]. Although they share some properties, there does not seem to be an obvious relation to Speyer's univariate polynomials introduced in [36]; examples Speyer1 and Speyer2 show they are relatively prime with their corresponding Ehrhart polynomials.…”

“…Since previous authors proposed other invariants of a matroid (e.g., the Tutte polynomials and the invariants of [8,36]) we wished to know how well the Ehrhart polynomial distinguishes nonisomorphic matroids. It is natural to compare it with other known invariants.…”

Symmetric and quasi-symmetric functions associated to polymatroids

“…Inspired by these graph invariants, Billera, Jia and Reiner defined a quasisymmetric function which is an invariant for matroids (see [3]). This invariant will be discussed later.…”