Delaunay polytopes derived from the Leech lattice

Mathieu Dutour Sikirić[1]; Konstantin Rybnikov[2]

  • [1] Rudjer Bosković Institute 54 ulica Bijenicka 1000 Zagreb, Croatia
  • [2] Department of Mathematical Sciences University of Massachusetts at Lowell MA 01854, Lowell, USA

Journal de Théorie des Nombres de Bordeaux (2014)

  • Volume: 26, Issue: 1, page 85-101
  • ISSN: 1246-7405

Abstract

top
A Delaunay polytope in a lattice 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 , which is higher than previously known .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 , which generalizes the Smith bound and we prove that this bound is met only by , the best known lattice covering in .

How to cite

top

Dutour Sikirić, Mathieu, and Rybnikov, Konstantin. "Delaunay polytopes derived from the Leech lattice." Journal de Théorie des Nombres de Bordeaux 26.1 (2014): 85-101. <http://eudml.org/doc/275734>.

@article{DutourSikirić2014,
abstract = {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 $\Lambda _\{24\}(v)^\{*\}$, which generalizes the Smith bound and we prove that this bound is met only by $\Lambda _\{23\}^\{*\}$, the best known lattice covering in $\mathbb\{R\}^\{23\}$.},
affiliation = {Rudjer Bosković Institute 54 ulica Bijenicka 1000 Zagreb, Croatia; Department of Mathematical Sciences University of Massachusetts at Lowell MA 01854, Lowell, USA},
author = {Dutour Sikirić, Mathieu, Rybnikov, Konstantin},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Delaunay polytopes; Leech lattice; covering radius; lamination number},
language = {eng},
month = {4},
number = {1},
pages = {85-101},
publisher = {Société Arithmétique de Bordeaux},
title = {Delaunay polytopes derived from the Leech lattice},
url = {http://eudml.org/doc/275734},
volume = {26},
year = {2014},
}

TY - JOUR
AU - Dutour Sikirić, Mathieu
AU - Rybnikov, Konstantin
TI - Delaunay polytopes derived from the Leech lattice
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2014/4//
PB - Société Arithmétique de Bordeaux
VL - 26
IS - 1
SP - 85
EP - 101
AB - 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 $\Lambda _{24}(v)^{*}$, which generalizes the Smith bound and we prove that this bound is met only by $\Lambda _{23}^{*}$, the best known lattice covering in $\mathbb{R}^{23}$.
LA - eng
KW - Delaunay polytopes; Leech lattice; covering radius; lamination number
UR - http://eudml.org/doc/275734
ER -

References

top
  1. A. Barvinok, A Course in Convexity. Graduate Studies in Mathematics 54, Amer. Math. Soc. 2002. Zbl1014.52001MR1940576
  2. H. Cohn, A. Kumar, Universally optimal distribution of points on spheres. J. Amer. Math. Soc. 20 (2007), 99–148. Zbl1198.52009MR2257398
  3. H. Cohn, A. Kumar, Optimality and uniqueness of the Leech lattice among lattices. Ann. of Math. 170 (2009), 1003–1050. Zbl1213.11144MR2600869
  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. Zbl0496.10020
  5. J. H. Conway, N. J. A. Sloane, Sphere Packings, Lattices and Groups (third edition). Grundlehren der mathematischen Wissenschaften 290, Springer–Verlag, 1999. Zbl0915.52003MR1662447
  6. H. S. M. Coxeter, Extreme forms. Canadian J. Math. 3 (1951), 391–441. Zbl0044.04201MR44580
  7. P. Delsarte, J. M. Goethals, J. J. Seidel, Spherical codes and designs. Geometriae Dedicata 6 (1977), 363–388. Zbl0376.05015MR485471
  8. M. Deza, M. Dutour, The hypermetric cone on seven vertices. Experiment. Math. 12 (2004), 433–440. Zbl1101.11021MR2043993
  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. Zbl0784.11027MR1218190
  10. M. Deza, M. Laurent, Geometry of cuts and metrics. Springer–Verlag, 1997. Zbl1210.52001MR1460488
  11. M. Dutour, Infinite serie of extreme Delaunay polytopes. European J. Combin. 26 (2005), 129–132. Zbl1062.52013MR2101040
  12. M. Dutour Sikirić, A. Schürmann, F. Vallentin, Complexity and algorithms for computing Voronoi cells of lattices. Math. Comp. 78 (2009), 1713–1731. Zbl1215.11067MR2501071
  13. M. Dutour Sikirić, V. Grishukhin, How to compute the rank of a Delaunay polytope. European J. Combin. 28 (2007) 762–773. Zbl1117.52015MR2300757
  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. Zbl1194.52018MR2342197
  16. M. Dutour Sikirić, A. Schürmann, F. Vallentin, Inhomogeneous extreme forms. Ann. Inst. Fourier 62 (2012), 2227–2255. Zbl1309.11057MR3060757
  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. Zbl1194.52018
  19. M. Dutour Sikirić, A. Schürmann, F. Vallentin, The contact polytope of the Leech lattice. Discrete Comput. Geom. 44 (2010), 904–911. Zbl1204.52015MR2728040
  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. Zbl0773.11042MR1176378
  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. Zbl1116.52006
  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. Zbl1162.52006
  24. R. Erdahl, K. Rybnikov, An infinite series of perfect quadratic forms and big Delaunay simplices in . Tr. Mat. Inst. Steklova 239 (2002), Diskret. Geom. i Geom. Chisel, 170–178; translation in Proc. Steklov Inst. Math. 239 (2002), 159–167. Zbl1126.11324
  25. V. Grishukhin, Infinite series of extreme Delaunay polytopes. European J. Combin. 27 (2006), 481–495. Zbl1088.52010MR2215210
  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. Zbl0255.50005MR307969
  28. J. Martinet, Perfect lattices in Euclidean spaces. Springer, 2003. Zbl1017.11031MR1957723
  29. R. E. O’Connor, G. Pall, The construction of integral quadratic forms of determinant . Duke Math. J. 11 (1944), 319–331. Zbl0060.11103MR10153
  30. W. Plesken, B. Souvignier, Computing isometries of lattices. J. Symbolic Comput. 24 (1997), 327–334. Zbl0882.11042MR1484483
  31. W. Plesken, Finite unimodular groups of prime degree and circulants. J. of Algebra 97 (1985), 286–312. Zbl0583.20036MR812182
  32. A. Schürmann, Experimental study of energy-minimizing point configurations on spheres. Amer. Math. Soc. Univ. Lect. Ser. 2009. Zbl1185.68771
  33. W. Smith, PhD thesis: studies in computational geometry motivated by mesh generation. Department of Applied Mathematics, Princeton University, 1988. MR2637686
  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. Zbl1139.11320MR1878745
  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. Zbl38.0261.01
  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/. 

NotesEmbed ?

top

You must be logged in to post comments.