Regular Two-graphs from the Even Unimodular Lattice E8⊕E8

AbstractStarting from the even unimodular latticeE8⊕E8,one constructs odd systems (i.e. sets of vectors with odd inner products) of 546 vectors using results of Deza and Grishukhin. One studies the subsystems consisting of 36 pairs of opposite vectors spanning equiangular lines. These subsystems represent regular two-graphs. This gives 100 such two-graphs and they coincide with the first 100 in a list of 227 two-graphs generated by E. Spence. Using the root systems of the sublattices generated by the 100 odd systems, the set of the 100 two-graphs is divided into seven classes. The first four classes correspond to the 23 Steiner triple system on 15 points containing a head, i.e. a Fano plane

The decomposition of the hypermetric cone into L-domains

The hypermetric cone \HYP_{n+1} is the parameter space of basic Delaunay polytopes in n-dimensional lattice. The cone \HYP_{n+1} is polyhedral; one way of seeing this is that modulo image by the covariance map \HYP_{n+1} is a finite union of L-domains, i.e., of parameter space of full Delaunay tessellations. In this paper, we study this partition of the hypermetric cone into L-domains. In particular, it is proved that the cone \HYP_{n+1} of hypermetrics on n+1 points contains exactly {1/2}n! principal L-domains. We give a detailed description of the decomposition of \HYP_{n+1} for n=2,3,4 and a computer result for n=5 (see Table \ref{TableDataHYPn}). Remarkable properties of the root system $\mathsf{D}_4$ are key for the decomposition of \HYP_5.Comment: 20 pages 2 figures, 2 table

On the sum of the Voronoi polytope of a lattice with 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? We give two necessary conditions and show that the vectors $U$ have to be free. Given a 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. In the case of the root lattice $\mathsf{E}_6$, 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. Based on a detailed study of the geometry of $P_V(\mathsf{e}_6)$, we give a simple characterization of the configurations of vectors $U$ such that $P_V(\mathsf{E}_6) + Z(U)$ is a parallelotope. The enumeration yields $10$ maximal families of vectors, which are presented by their description as regular matroids.Comment: 30 pages, 4 figures, 4 table

Regular Two-Graphs From the Even Unimodular Lattice ...

This is a revised version of the first part of [8]. Each two-graph is in one-to-one correspondence with a set of equiangular lines. This implies that a two-graph is represented by a system of vectors of equal odd norm with mutual inner products \Sigma1. This is a special odd system, i.e. a set of vectors with odd inner products. The construction of obtaining odd systems from doubly even lattices introduced in [5], [6] is applied to the even unimodular lattice E 8 \Phi E 8 multiplied by p 2. For the odd system of 456 vectors obtained by this construction, we study odd subsystems of 36 vectors spanning equiangular lines, i.e. subsystems representing regular two-graphs on 36 points. A subsystem of vectors representing a two-graph generates a sublattice of the lattice E 8 \Phi E 8 . These sublattices are distinguished by sets of lattice vectors of norm 2. These sets are root systems. Hence the set of all two-graphs from E 8 \Phi E 8 is partitioned into families of two-graphs with the s..

Combinatorics of Delaunay polytopes of the isodual lattice Q 10

The results of [9] are generalized and simplified for code lattices. As an example, the code lattice Q 10 , mentioned and named in the paper [6], is considered. Q 10 has two symmetric Delaunay polytopes P 5 , P 3 and an asymmetric P 0 5 , and is generated by P 5 . P 5 is a symmetrization of the cut polytope PCut 5 , i.e. it is the convex hull of all cuts and their complements in the complete graph K 5 . The cuts and their complements are all circuits of the regular matroid R 10 [12]. Besides P 5 is the convex hull of the unique 10-dimensional closed odd system of 16 pairs of opposite vectors of norm 5 spanning equiangular lines at angle arccos 1 5 . P 5 is also the convex hull of all codewords of the linear binary code C 10 = [10; 5; 4]. The second symmetric Delaunay polytopes P 3 is the convex hull of a system of 40 pairs of opposite vectors of norm 3 with 0; \Sigma1 inner products. The asymmetric Delaunay polytope P 0 5 is the convex hull of an odd system of 32 vectors of norm ..