The Minkowski sum of a zonotope and the Voronoi polytope of the root lattice E7

摘要

We show that the Minkowski sum PV(E7) + Z(U) of the Voronoi polytope PV(E7) of the root lattice E7 and the zonotope Z(U) is a 7-dimensional parallelohedron if and only if the set U consists of minimal vectors of the dual lattice E7 up to scalar multiplication, and U does not contain forbidden sets. The minimal vectors of E7 are the vectors r of the classical root system E7. If the r2-norm of the roots is set equal to 2, then the scalar products of minimal vectors from the dual lattice only take the values ±1/2. A set of minimal vectors is referred to as forbidden if it consists of six vectors, and the directions of some of these vectors can be changed so as to obtain a set of six vectors with all the pairwise scalar products equal to 1/2.