Delaunay polytopes derived from the Leech lattice

Dutour Sikirić, Mathieu; Rybnikov, Konstantin

Journal de théorie des nombres de Bordeaux, Tome 26 (2014) no. 1, pp. 85-101.

Un polytope de Delaunay dans un réseau $L$ est parfait si les transformations affine qui préservent sa propriété d’être Delaunay sont des compositions d’homothéties et d’isométries. Les polytopes de Delaunay parfaits sont rares en petite dimension et ici nous considérons ceux qui apparaissent dans des sections du réseau de Leech.

Par ce moyen, nous trouvons des rśeaux ayant plusieurs orbites de polytopes de Delaunay parfaits. Nous exhibons aussi des polytopes de Delaunay parfait qui restent Delaunay dans des super-réseaux. Aussi nous trouvons des polytopes de Delaunay parfait ayant des groupes d’automorphismes relativement petit par rapport à leurs réseaux. Nous prouvons aussi que certains polytopes de Delaunay parfaits ont un nombre de lamination égal à $5$ alors que les polytopes de Delaunay précédemment connus ont un nombre de lamination égal à $3$.

Une construction bien connu de polytopes de Delaunay centralement symmétriques utilise des polytope de Delaunay parfait antisymmétriques. Nous classifions complètement les types de polytopes de Delaunay parfaits qui peuvent apparaîtrent dans cette construction.

Enfin, nous prouvons une borne supérieure sur le rayon de recouvrement du réseau ${\Lambda }_{24}{\left(v\right)}^{*}$ qui généralise la borne de Smith. Nous prouvons que cette borne est atteinte seulement pour ${\Lambda }_{23}^{*}$ qui est le meilleur recouvrement de ${ℝ}^{23}$ connu.

A Delaunay polytope in a lattice $L$ is perfect if any affine transformation that preserve its Delaunay property is a composite of an homothety and an isometry. Perfect Delaunay polytopes are rare in low dimension and here we consider the ones that one can get in lattice that are sections of the Leech lattice.

By doing so we are able to find lattices with several orbits of perfect Delaunay polytopes. Also we exhibit Delaunay polytopes which remain Delaunay in some superlattices. We found perfect Delaunay polytopes with small automorphism group relative to the automorphism group of the lattice. And we prove that some perfect Delaunay polytopes have lamination number $5$, which is higher than previously known $3$.

A well known construction of centrally symmetric perfect Delaunay polytopes uses a laminated construction from an antisymmetric perfect Delaunay polytope. We fully classify the types of perfect Delaunay polytopes that can occur.

Finally, we derived an upper bound for the covering radius of ${\Lambda }_{24}{\left(v\right)}^{*}$, which generalizes the Smith bound and we prove that this bound is met only by ${\Lambda }_{23}^{*}$, the best known lattice covering in ${ℝ}^{23}$.

Classification : 11H31, 11H55

[1] A. Barvinok, A Course in Convexity. Graduate Studies in Mathematics 54, Amer. Math. Soc. 2002. | MR | Zbl

[2] H. Cohn, A. Kumar, Universally optimal distribution of points on spheres. J. Amer. Math. Soc. 20 (2007), 99–148. | MR | Zbl

[3] H. Cohn, A. Kumar, Optimality and uniqueness of the Leech lattice among lattices. Ann. of Math. 170 (2009), 1003–1050. | MR | Zbl

[4] J. H. Conway, R. A. Parker, N. J. A. Sloane, The covering radius of the Leech lattice. Proc. Roy. Soc. London Ser. A 380 (1982), 261–290. | Zbl

[5] J. H. Conway, N. J. A. Sloane, Sphere Packings, Lattices and Groups (third edition). Grundlehren der mathematischen Wissenschaften 290, Springer–Verlag, 1999. | MR | Zbl

[6] H. S. M. Coxeter, Extreme forms. Canadian J. Math. 3 (1951), 391–441. | MR | Zbl

[7] P. Delsarte, J. M. Goethals, J. J. Seidel, Spherical codes and designs. Geometriae Dedicata 6 (1977), 363–388. | MR | Zbl

[8] M. Deza, M. Dutour, The hypermetric cone on seven vertices. Experiment. Math. 12 (2004), 433–440. | MR | Zbl

[9] M. Deza, V. P. Grishukhin, M. Laurent, Extreme hypermetrics and L-polytopes. In G. Halász et al. eds, Sets, Graphs and Numbers, Budapest (Hungary), 1991, 60 Colloquia Mathematica Societatis János Bolyai (1992), 157–209. | MR | Zbl

[10] M. Deza, M. Laurent, Geometry of cuts and metrics. Springer–Verlag, 1997. | MR | Zbl

[11] M. Dutour, Infinite serie of extreme Delaunay polytopes. European J. Combin. 26 (2005), 129–132. | MR | Zbl

[12] M. Dutour Sikirić, A. Schürmann, F. Vallentin, Complexity and algorithms for computing Voronoi cells of lattices. Math. Comp. 78 (2009), 1713–1731. | MR | Zbl

[13] M. Dutour Sikirić, V. Grishukhin, How to compute the rank of a Delaunay polytope. European J. Combin. 28 (2007) 762–773. | MR | Zbl

[14] M. Dutour Sikirić, K. Rybnikov, Perfect but not generating Delaunay polytopes. Special issue of Symmetry Culture and Science on tesselations, Part II, 317–326.

[15] M. Dutour Sikirić, R. Erdahl, K. Rybnikov, Perfect Delaunay polytopes in low dimensions. Integers 7 (2007) A39. | MR | Zbl

[16] M. Dutour Sikirić, A. Schürmann, F. Vallentin, Inhomogeneous extreme forms. Ann. Inst. Fourier 62 (2012), 2227–2255. | Numdam | MR

[17] M. Dutour Sikirić, Enumeration of inhomogeneous perfect forms. In preparation.

[18] M. Dutour Sikirić, K. Rybnikov, A new algorithm in geometry of numbers. In Proceedings of ISVD-07, the 4-th International Symposium on Voronoi Diagrams in Science and Engineering, Pontypridd, Wales, July 2007. IEEE Publishing Services, 2007.

[19] M. Dutour Sikirić, A. Schürmann, F. Vallentin, The contact polytope of the Leech lattice. Discrete Comput. Geom. 44 (2010), 904–911. | MR | Zbl

[20] R. Erdahl, A convex set of second-order inhomogeneous polynomials with applications to quantum mechanical many body theory. Mathematical Preprint #1975-40, Queen’s University, Kingston, Ontario, 1975.

[21] R. Erdahl, A cone of inhomogeneous second-order polynomials. Discrete Comput. Geom. 8 (1992), 387–416. | MR | Zbl

[22] R. M. Erdahl, K. Rybnikov, Voronoi-Dickson Hypothesis on Perfect Forms and L-types. Peter Gruber Festshrift: Rendiconti del Circolo Matematiko di Palermo, Serie II, Tomo LII, part I (2002), 279–296. | Zbl

[23] R.M. Erdahl, A. Ordine, K. Rybnikov, Perfect Delaunay Polytopes and Perfect Quadratic Functions on Lattices. Integer points in polyhedra—geometry, number theory, representation theory, algebra, optimization, statistics, Contemporary Mathematics 452, American Mathematical Society, 2008, 93–114. | Zbl

[24] R. Erdahl, K. Rybnikov, An infinite series of perfect quadratic forms and big Delaunay simplices in ${\mathbb{Z}}^n$. Tr. Mat. Inst. Steklova 239 (2002), Diskret. Geom. i Geom. Chisel, 170–178; translation in Proc. Steklov Inst. Math. 239 (2002), 159–167. | Zbl

[25] V. Grishukhin, Infinite series of extreme Delaunay polytopes. European J. Combin. 27 (2006), 481–495. | MR | Zbl

[26] J.-M. Kantor, Lattice polytope: some open problems. AMS Snowbird Proceedings.

[27] P.W. Lemmens, J.J. Seidel, Equiangular lines. J. Algebra 24 (1973), 494–512. | MR | Zbl

[28] J. Martinet, Perfect lattices in Euclidean spaces. Springer, 2003. | MR | Zbl

[29] R. E. O’Connor, G. Pall, The construction of integral quadratic forms of determinant $1$. Duke Math. J. 11 (1944), 319–331. | MR | Zbl

[30] W. Plesken, B. Souvignier, Computing isometries of lattices. J. Symbolic Comput. 24 (1997), 327–334. | MR | Zbl

[31] W. Plesken, Finite unimodular groups of prime degree and circulants. J. of Algebra 97 (1985), 286–312. | MR | Zbl

[32] A. Schürmann, Experimental study of energy-minimizing point configurations on spheres. Amer. Math. Soc. Univ. Lect. Ser. 2009. | Zbl

[33] W. Smith, PhD thesis: studies in computational geometry motivated by mesh generation. Department of Applied Mathematics, Princeton University, 1988. | MR

[34] B. B. Venkov, Réseaux et designs sphériques. In Réseaux euclidiens, designs sphériques et formes modulaires, edited by J. Martinet, Monographie numéro 37 de L’enseignement Mathématique, 2001. | MR | Zbl

[35] G. F. Voronoi, Nouvelles applications des paramètres continus à la théorie des formes quadratiques - Deuxième mémoire. J. für die Reine und Angewandte Mathematik, 134 (1908), 198-287 and 136 (1909), 67–178.

[36] N. J. A. Sloane, G. Nebe, A Catalogue of Lattices. http://www2.research.att.com/~njas/lattices/. Software

[37] M. Dutour Sikirić, polyhedral, http://www.liga.ens.fr/~dutour/Polyhedral/.