Perfect lattice
{{short description|Euclidean 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 {{harvtxt|Korkine|Zolotareff|1877}}. A strongly perfect lattice is one whose minimal vectors form a spherical 4-design. This notion was introduced by {{harvtxt|Venkov|2001}}.
{{harvtxt|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 {{OEIS|id=A004026}}. {{harvtxt|Conway|Sloane|1988}} summarize the properties of perfect lattices of dimension up to 7.
{{harvtxt|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 {{harvtxt|Riener|2006}} that only 2408 of these 10916 perfect lattices in dimension 8 are actually extreme lattices.
References
- {{Citation | last1=Conway | first1=John Horton | author1-link=John Horton Conway | last2=Sloane | first2=N. J. A. | author2-link=Neil Sloane | title=Low-dimensional lattices. III. Perfect forms | jstor=2398316 | mr=953277 | year=1988 | journal=Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences | issn=0962-8444 | volume=418 | issue=1854 | pages=43–80 | doi=10.1098/rspa.1988.0073| bibcode=1988RSPSA.418...43C }}
- {{cite journal |title = Errata: Low-Dimensional Lattices. III. Perfect Forms| journal = Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences | jstor = 2398351 | volume=426| issue = 1871 | pages = 441 | last1 = Conway | first1 = J. H. | last2 = Sloane | first2 = N. J. A. | year = 1989 | doi = 10.1098/rspa.1989.0134 | bibcode = 1989RSPSA.426..441C }}
- {{Citation | last1=Korkine | last2=Zolotareff | title=Sur les formes quadratique positives | doi=10.1007/BF01442667 | year=1877 | journal=Mathematische Annalen | issn=0025-5831 | volume=11 | issue=2 | pages=242–292| url=https://zenodo.org/record/1896288 }}
- {{Citation | last1=Martinet | first1=Jacques | title=Perfect lattices in Euclidean spaces | url=https://books.google.com/books?id=gd9CcFclBRIC | publisher=Springer-Verlag | location=Berlin, New York | series=Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] | isbn=978-3-540-44236-3 | mr=1957723 | year=2003 | volume=327| doi=10.1007/978-3-662-05167-2 | url-access=subscription }}
- {{Citation | last1=Riener | first1=Cordian | title=On extreme forms in dimension 8 | url=https://eudml.org/doc/249637 | journal= Journal de théorie des nombres de Bordeaux | volume=18 | issue=3 |pages=677–682 | year=2006 | doi=10.5802/jtnb.565| doi-access=free }}
- {{Citation | last1=Sikirić | first1=Mathieu Dutour | last2=Schürmann | first2=Achill | last3=Vallentin | first3=Frank | title=Classification of eight-dimensional perfect forms | arxiv=math/0609388 | doi=10.1090/S1079-6762-07-00171-0 | mr=2300003 | year=2007 | journal=Electronic Research Announcements of the American Mathematical Society | issn=1079-6762 | volume=13 | issue=3 | pages=21–32}}
- {{Citation |last1=Venkov |first1=Boris|title=Réseaux et designs sphériques, Réseaux euclidiens, designs sphériques et formes modulaires| journal= Monographie de l'Enseignement Mathématique |volume=37 |year=2001|pages= 10–86}}
- {{Citation | last1=Voronoi | first1=G. | title=Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Premier Mémoire: Sur quelques propriétés des formes quadratiques positives parfaites | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002166534 | language=French | doi=10.1515/crll.1908.133.97 | year=1908 | journal=Journal für die reine und angewandte Mathematik | issn=0075-4102 | volume=1908 | issue=133 | pages=97–178| url-access=subscription }}
External links
- [http://www.math.uni-magdeburg.de/lattice_geometry/ List of perfect lattices in dimension 8]