Symmetrizing Polytopes

A classic operation in combinatorics is the composition of two structures. For instance, an ordered set partition can be viewed as the composition of a set partition and a permutation. Geometrically, we can view this composition through the permutohedron, which realizes ordered set partitions as a polytope. Motivated by this viewpoint, we explore how to compose polytopes that carry rich combinatorial structures. In this talk, I will present a construction for the case when one of the polytopes is a permutohedron. The key idea is to use a finite reflection group G to symmetrize an arbitrary polytope P, and to show that the combinatorics of the resulting polytope G(P) can be recovered from the intersection of the normal fan of P with the fundamental domain of G.

This is joint work with Federico Castillo.