On the sum of a parallelotope and a zonotope

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 ( L ) 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 ( U ) , 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 ( U ) 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 ( U ) is a parallelotope. Using this we classify such zonotopes for some highly symmetric lattices. 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 ( E 6 ) , we give a simple characterization of the configurations of vectors U such that P V ( E 6 ) + Z ( U ) is a parallelotope. The enumeration yields 10 maximal families of vectors, which are presented by their description as regular matroids.