On Certain Coxeter Lattices Without Perfect Sections
Anne-Marie Bergé
DOI: 10.1023/B:JACO.0000047289.61038.1f
Abstract
In this paper, we compute the kissing numbers of the sections of the Coxeter lattices \mathbb A n \frac n\text + 12 {\mathbb{A}}_n ^{\frac{{n{\text{ + }}1}}{2}} , n odd, and in particular we prove that for n 
7 they cannot be perfect. The proof is merely combinatorial and relies on the structure of graphs canonically attached to the sections.
7 they cannot be perfect. The proof is merely combinatorial and relies on the structure of graphs canonically attached to the sections.Pages: 5–16
Keywords: perfect lattice; kissing number; bipartite graph
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References
1 C. Batut and J. Martinet, “Radiographie des réseaux parfaits”, Experimental Math. 3 (1994), 39-49. 2 B. Bollobás, Modern Graph Theory, Graduate texts in Mathematics 184, Springer-Verlag, Heidelberg, 1998. 3 A.M. Bergé and J. Martinet, “ Symmetric groups and lattices”, Monatschefte f\ddot ur Math. 140(3) (2003), 179-195. 4 J.H. Conway and N.J.A. Sloane, “Low-dimensional lattices. III. Perfect forms”, Proc. Royal Soc. London A 418 (1988), 43-80. 5 J. Martinet, personal communication. 6 J. Martinet, Perfect Lattices in Euclidean Spaces, Grundlehren 327, Springer-Verlag, Heidelberg, 2003. 7 J. Martinet and B. Venkov, “On integral lattices having an odd minimum,” Algebra and Analysis, Saint- Petersburg 16(3) (2004), 198-237.