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 (1877). 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 (1988) summarize the properties of perfect lattices of dimension up to 7. Sikirić, Schürmann & Vallentin (2007) 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,
Bibcode:
1988RSPSA.418...43C,
doi:
10.1098/rspa.1988.0073,
ISSN
0962-8444,
JSTOR
2398316,
MR
0953277
- 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. Bibcode: 1989RSPSA.426..441C. doi: 10.1098/rspa.1989.0134. JSTOR 2398351.
- Korkine; Zolotareff (1877), "Sur les formes quadratique positives", Mathematische Annalen, 11 (2): 242–292, doi: 10.1007/BF01442667, ISSN 0025-5831
- Martinet, Jacques (2003), Perfect lattices in Euclidean spaces, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 327, Berlin, New York: Springer-Verlag, doi: 10.1007/978-3-662-05167-2, ISBN 978-3-540-44236-3, MR 1957723
- 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
- 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, arXiv: math/0609388, doi: 10.1090/S1079-6762-07-00171-0, ISSN 1079-6762, MR 2300003
- 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", Journal für die reine und angewandte Mathematik (in French), 1908 (133): 97–178, doi: 10.1515/crll.1908.133.97, ISSN 0075-4102
External links
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 (1877). 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 (1988) summarize the properties of perfect lattices of dimension up to 7. Sikirić, Schürmann & Vallentin (2007) 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,
Bibcode:
1988RSPSA.418...43C,
doi:
10.1098/rspa.1988.0073,
ISSN
0962-8444,
JSTOR
2398316,
MR
0953277
- 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. Bibcode: 1989RSPSA.426..441C. doi: 10.1098/rspa.1989.0134. JSTOR 2398351.
- Korkine; Zolotareff (1877), "Sur les formes quadratique positives", Mathematische Annalen, 11 (2): 242–292, doi: 10.1007/BF01442667, ISSN 0025-5831
- Martinet, Jacques (2003), Perfect lattices in Euclidean spaces, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 327, Berlin, New York: Springer-Verlag, doi: 10.1007/978-3-662-05167-2, ISBN 978-3-540-44236-3, MR 1957723
- 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
- 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, arXiv: math/0609388, doi: 10.1090/S1079-6762-07-00171-0, ISSN 1079-6762, MR 2300003
- 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", Journal für die reine und angewandte Mathematik (in French), 1908 (133): 97–178, doi: 10.1515/crll.1908.133.97, ISSN 0075-4102