On the sum of a parallelotope and a zonotope

Abstract

A parallelotope $P$ is a polytope that admits a facet-to-facet tiling of space by translation copies of $P$ along a lattice. The Voronoi cell $P_V\left(L\right)$ of a lattice $L$ is an example of a parallelotope. A parallelotope can be uniquely decomposed as the Minkowski sum of a zone closed parallelotope $P$ and a zonotope $Z\left(U\right)$, where $U$ is the set of vectors used to generate the zonotope. In this paper we consider the related question: When is the Minkowski sum of a general parallelotope and a zonotope $P+Z\left(U\right)$ a parallelotope? Two necessary conditions are that the vectors of $U$ have to be free and form a unimodular set. Given a unimodular set $U$ of free vectors, we give several methods for checking if $P+Z\left(U\right)$ is a parallelotope. Using this we classify such zonotopes for some highly symmetric lattices.

In the case of the root lattice $E_6$, it is possible to give a more geometric description of the admissible sets of vectors $U$. We found that the set of admissible vectors, called free vectors, is described by the well-known configuration of 27 lines in a cubic surface. Based on a detailed study of the geometry of $P_V\left(E_6\right)$, we give a simple characterization of the configurations of vectors $U$ such that $P_V\left(E_6\right)+Z\left(U\right)$ is a parallelotope. The enumeration yields 10 maximal families of vectors, which are presented by their description as regular matroids.