Perfect lattice
In mathematics, a perfect lattice (or perfect form) is a lattice in a Euclidean vector space, that is completely determined by the set S of its minimal vectors in the sense that there is only one positive definite quadratic form taking value 1 at all points of S. Perfect lattices were introduced by (Korkine Zolotareff). A strongly perfect lattice is one whose minimal vectors form a spherical 4-design. This notion was introduced by (Venkov 2001).
(Voronoi 1908) proved that a lattice is extreme if and only if it is both perfect and eutactic.
The number of perfect lattices in dimensions 1, 2, 3, 4, 5, 6, 7, 8 is given by 1, 1, 1, 2, 3, 7, 33, 10916 (sequence A004026 in the OEIS). (Conway Sloane) summarize the properties of perfect lattices of dimension up to 7. (Sikirić Schürmann) verified that the list of 10916 perfect lattices in dimension 8 found by Martinet and others is complete. It was proven by (Riener 2006) that only 2408 of these 10916 perfect lattices in dimension 8 are actually extreme lattices.
References
- Conway, John Horton; Sloane, N. J. A. (1988), "Low-dimensional lattices. III. Perfect forms", Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 418 (1854): 43–80, doi:10.1098/rspa.1988.0073, ISSN 0962-8444, Bibcode: 1988RSPSA.418...43C
- Conway, J. H.; Sloane, N. J. A. (1989). "Errata: Low-Dimensional Lattices. III. Perfect Forms". Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 426 (1871): 441. doi:10.1098/rspa.1989.0134. Bibcode: 1989RSPSA.426..441C.
- Korkine; Zolotareff (1877), "Sur les formes quadratique positives", Mathematische Annalen 11 (2): 242–292, doi:10.1007/BF01442667, ISSN 0025-5831, https://zenodo.org/record/1896288
- Martinet, Jacques (2003), Perfect lattices in Euclidean spaces, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 327, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-662-05167-2, ISBN 978-3-540-44236-3, https://books.google.com/books?id=gd9CcFclBRIC
- Riener, Cordian (2006), "On extreme forms in dimension 8", Journal de théorie des nombres de Bordeaux 18 (3): 677–682, doi:10.5802/jtnb.565, https://eudml.org/doc/249637
- Sikirić, Mathieu Dutour; Schürmann, Achill; Vallentin, Frank (2007), "Classification of eight-dimensional perfect forms", Electronic Research Announcements of the American Mathematical Society 13 (3): 21–32, doi:10.1090/S1079-6762-07-00171-0, ISSN 1079-6762
- Venkov, Boris (2001), "Réseaux et designs sphériques, Réseaux euclidiens, designs sphériques et formes modulaires", Monographie de l'Enseignement Mathématique 37: 10–86
- Voronoi, G. (1908), "Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Premier Mémoire: Sur quelques propriétés des formes quadratiques positives parfaites" (in French), Journal für die reine und angewandte Mathematik 1908 (133): 97–178, doi:10.1515/crll.1908.133.97, ISSN 0075-4102, http://resolver.sub.uni-goettingen.de/purl?GDZPPN002166534
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