Voronoi Domains and Dual Cells in the Generalized Kaleidoscope with Applications to Root and Weight Lattices

Abstract

AbstractWe give a uniform description, in terms of Coxeter diagrams, of the Voronoi domains of the root and weight lattices of any semisimple Lie algebra. This description provides a classification not only of all the facets of these Voronoi domains but simultaneously a classification of their dual or Delaunay cells and their facets. It is based on a much more general theory that we develop here providing the same sort of information in the setting of chamber geometries defined by arbitrary reflection groups. These generalized kaleidoscopes include the classical spherical, Euclidean, and hyperbolic kaleidoscopes as special cases. We prove that under certain conditions the Delaunay cells are Voronoi cells for the vertices of the Voronoi complex. This leads to the description in terms of Wythoff polytopes of the Voronoi cells of the weight lattices.