The University of Michigan Combinatorics Seminar
Fall 2006
October 20, 4:10-5:00, 3866 East Hall



Decompositions of matroid polytopes
and a new matroid invariant

David Speyer

Clay Institute and the University of Michigan


Abstract

For M a rank d matroid on {1,...,n}, let P_M be the convex hull of the vectors e_{i_1}+...+e_{i_d}, where (i_1, ..., i_d) ranges over the bases of M. We will call a polytope of the form P_M matroidal. We will study decompositions of matroidal polytopes into smaller matroidal polytopes. Such decompositions arise in algebraic geometry in the study of tropical linear spaces or of chow quotients of grassmannians; they are also natural from a purely combinatorial standpoint because matroid invariants such as the Tutte polynomial are additive under such decompositions. I previously conjectured that any such decomposition uses at most (n-c-1)!/(d-c)!(n-d-c)!(c-1)! internal faces of dimension n-c. I have now found a proof in the case that all matroids involved are realizable over C. In the process, I have discovered a matroid invariant which is additive under the sort of polyhedral decompositions described above and which behaves simply under direct sum, duality, series-parallel extension and two sum. I will define this invariant, describe its properties and give a simple description of it in the case of transversal matroids. The motivation for my definition comes from the equivariant K-theory of the grassmannian; a simple combinatorial definition is still lacking.