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Kurt Luoto

A matroid-friendly basis for quasisymmetric functions.

Abstract for the Combinatorics Seminar 2007 September 11

Matroid base polytope decompositions arise in the work of certain algebraic geometers. In 2006, Billera, Jia, and Reiner invented a new invariant F(M) for matroids in the form of a quasisymmetric function. One motivating application of this invariant is to the study of matroid base polytope decompositions. The mapping of matroids to the algebra of quasisymmetric functions (QSym) behaves as a valuation on matroid base polytopes, and leads to a necessary algebraic condition on their decompositions. Billera, Jia, and Reiner pose a number of questions regarding this relationship. We address some of these questions, obtaining a full characterization for the rank two case.

Along the way, we obtain a novel Z-basis for QSym that has especially nice properties. For instance this basis has nonnegative integer structure constants and reflects, in addition to the usual grading of QSym by degree, a second grading of QSym that on (the images of) loopless matroids coincides with their matroidal rank.

No familiarity with quasisymmetric functions or matroids is assumed for this talk.

seminars/comb/abstract.200709luo.txt · Last modified: 2020/01/29 14:03 (external edit)