The University of Michigan Combinatorics Seminar
Some known invariants for matroids are the classical Tutte polynomial, the quasi-symmetric function F of Billera, Jia and Reiner, another quasi-symmetric function G introduced by the speaker, and an invariant introduced by Speyer. All of these invariants have the valuative property (in the case of Speyer's invariant, after a "twist"). This means that the invariants behave nicely with respect to decompositions of matroid polytopes into smaller matroid polytopes. For fixed rank and cardinality, I will describe a basis for all the valuative functions and valuative invariants for a matroid. This will imply that the quasi-symmetric function G is a universal invariant: it specializes to every valuative invariant. The results will also be generalized to polymatroids. This is joint work with Alex Fink.