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

Moody, R. V. ; Patera, J.

Canadian journal of mathematics, Tome 47 (1995) no. 3, pp. 573-605

Résumé

We 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.

Détail

DOI : 10.4153/CJM-1995-031-2

Mots-clés : 70F35, 70F10, 70F15, 58F40, Voronoi cell, Delaunay cell, generalized kaleidoscope, root lattice, weight lattice, Coxeter group, Wythoff polytope

Bibliographie

[B] Bourbaki, N., Groupes et algèbres de Lie, Chapter 4,5,6, Éléments de mathématiques, Hermann, Paris, 1968. Google Scholar

[BMP] Bremner, M.R., Moody, R.V. and Patera, J., Tables of dominant weight multiplicities for representations of simple Lie algebras, Marcel Dekker, Inc., New York, 1985. Google Scholar

[Br] Brondsted, A., An introduction to convex polytopes. In: Graduate Texts in Math., Springer-Verlag, New York, 1983. Google Scholar

[Cl] Coxeter, H.S.M., Regular polytopes, Dover, New York, 1973. Google Scholar

[C2] Coxeter, H.S.M., Generators and relations for discrete groups, 2nd ed, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, New York, 1965. Google Scholar

[CS1] Conway, J. and Sloane, N., Sphere packing, lattices, and groups, Springer-Verlag, New York, 1988. Google Scholar

[CS2] Conway, J., The cell structures of certain lattices, In: Miscelánea Mat., (eds. Hilton, P., Hirzebruch, F., and Remmert, R.), Springer-Verlag, New York, 1991. 71–107. Google Scholar

[KPZ] Kramer, P., Papadopolos, Z. and Zeidler, D., The root lattice D6 and icosahedral quasicrystals, Proc. Symp. Symmetries in Physics, Cocoyoc, Mexico 1991, Lecture Notes in Physics, (eds. Frank, A. and Wolf, K.B.), Springer-Verlag, New York, 1992. Google Scholar

[Ma] Maxwell, G., Wythoff's construction for Coxeter groups, J. Algebra 123(1989). Google Scholar

[MPI] Moody, R.V. and Patera, J., Voronoi and Delaunay cells of root lattices: Classification of their faces and facets by Coxeter-Dynkin diagrams, J. Phys. A, Math. Gen. 25(1992), 5089–5134. Google Scholar

[MP2] Moody, R.V., Quasicrystals and icosions, J. Phys. A, Math. Gen. 26(1993), 2829–2852. Google Scholar

[Sh] Scharlau, R., Geometrical realizations of shadow geometries, Proc. London Math. Soc. 61, 615–656. Google Scholar

[Se] Senechal, M., Introduction to lattice geometry. In: From number theory to physics, (eds. Moussa, P. and Waldschmidt, M.), Springer-Verlag, New York, to appear. Google Scholar

[TBV] Tits, J., Bratslavski, F. and Valette, G., Algèbres de Lie semi-simples et systèmes de racines, In: Lecture Notes, Centre Belge d'algèbre et de topologie, Université Libre de Bruxelles, 1964. Google Scholar

[Wl] Worley, R.T., The Voronoi region of E*6, J. Austral. Math. Soc. Ser. A 48(1987), 268–278. Google Scholar

[W2] Worley, R.T., The Voronoi region of E*v, SIAM J. Math.Discrete 1(1988), 134–141. Google Scholar