TY - JOUR

T1 - A quasisymmetric function for matroids

AU - Billera, Louis J.

AU - Jia, Ning

AU - Reiner, Victor

PY - 2009/11/1

Y1 - 2009/11/1

N2 - 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.

AB - 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.

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U2 - 10.1016/j.ejc.2008.12.007

DO - 10.1016/j.ejc.2008.12.007

M3 - Article

AN - SCOPUS:70349304326

VL - 30

SP - 1727

EP - 1757

JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

SN - 0195-6698

IS - 8

ER -