Summary: Valuations for matroid polytope subdivisions.
We prove that the ranks of the subsets and the activities of the
bases of a matroid define valuations for the subdivisions of a matroid
polytope into smaller matroid polytopes.
Aside from its wide applicability in many areas of mathematics, one of the
pleasant features of matroid theory is the availability of a vast number of
equivalent points of view. Among many others, one can think of a matroid
as a notion of independence, a closure relation, or a lattice. One point of
view has gained prominence due to its applications in algebraic geometry,
combinatorial optimization, and Coxeter group theory: that of a matroid
as a polytope. This paper is devoted to the study of functions of a matroid
which are amenable to this point of view.
To each matroid M one can associate a (basis) matroid polytope Q(M),
which is the convex hull of the indicator vectors of the bases of M. One can
recover M from Q(M), and in certain instances Q(M) is the fundamental