Combinatorial aspects of polytope slices

Abstract

We studies two examples of polytope slices, hypersimplices as slices of hypercubes and edge polytopes. For hypersimplices, the main result is a proof of a conjecture by R. Stanley which gives an interpretation of the Ehrhart h*-vector in terms of descents and excedances. Our proof is geometric using a careful book-keeping of a shelling of a unimodular triangulation. We generalize this result to other closely related polytopes. We next study slices of edge polytopes. Let G be a finite connected simple graph with d vertices and let PG C Rd be the edge polytope of G. We call PG decomposable if PG decomposes into integral polytopes PG+ and PG- via a hyperplane, and we give an algorithm which determines the decomposability of an edge polytope. Based on a sequence of papers by Ohsugi and Hibi, we prove that when PG is decomposable, PG is normal if and only if both PG+ and PG- are normal. We also study toric ideals of PG, PG+ and PG-. This part is joint work with Hibi and Zhang.