Voronoi and Delaunay cells of root lattices: classification of their faces and facets by Coxeter-Dynkin diagrams

The Voronoi domains, their duals (Delaunay domains) and all their faces of any dimension are classified and described in terms of the Weyl group action on a representative of each type of face. The representative of a face type is specified by a decoration of the corresponding Coxeter-Dynkin diagram. The rules of domain description are uniform for root lattices of simple Lie groups of all types. An explicit description of the representatives of all faces is carried out for the domains of root lattices of the four classical series and for the five exceptional simple Lie groups. The Coxeter-Dynkin diagrams required here are the diagrams extended by the highest short root. Each diagram is partitioned into two subdiagrams, one describing completely a d-face of the Voronoi domain, its complement completely describing the dual of the d-face. The applicability of the authors' classification method to generalized kaleidoscopes is explained.