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Encyclopedia > Coxeter number

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Niemeier lattice In mathematics, a Niemeier lattice is one of the 24 positive definite even unimodular lattices of rank 24, which were classified by Niemeier. The best known example is the Leech lattice.



Classification Niemeier lattices are usually labeled by the Dynkin diagram of their root systems. These Dynkin diagrams have rank either 0 or 24, and all of their components have the same Coxeter number. (The Coxeter number, at least in these cases, is the number of roots divided by the dimension.) There are exactly 24 Dynkin diagrams with these properties, and there turns out to be a unique Niemeier lattice for each of these Dynkin diagrams.


The complete list of Niemeier lattices is


The Leech lattice (empty root system), Coxeter number 0. A124, Coxeter number 2. A212, Coxeter number 3. A38, Coxeter number 4. A46, Coxeter number 5. A54D4, Coxeter number 6. D46, Coxeter number 6. A64, Coxeter number 7. A72D52, Coxeter number 8. A83, Coxeter number 9. A92D6, Coxeter number 10. D64, Coxeter number 10. E64, Coxeter number 12. A11D7E6, Coxeter number 12. A122, Coxeter number 13. D83, Coxeter number 14. A15D9, Coxeter number 16. A17E7, Coxeter number 18. D10E72, Coxeter number 18. D122, Coxeter number 22. A24, Coxeter number 25. D16E8, Coxeter number 30. E83, Coxeter number 30. D24, Coxeter number 46.


  Results from FactBites:
 
Coxeter group - Wikipedia, the free encyclopedia (856 words)
In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries.
Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example.
Examples of infinite Coxeter groups include the triangle groups corresponding to regular tesselations of the Euclidean plane and the hyperbolic plane, and the Weyl groups of infinite-dimensional Kac–Moody algebras.
Coxeter number (208 words)
In mathematics, the Coxeter number h is the order of a Coxeter element of an irreducible root system, Weyl group, or Coxeter group.
The Coxeter number is the number of roots divided by the rank.
The Coxeter number is the highest degree of a fundamental invariant of the Weyl group acting on polynomials.
  More results at FactBites »


 

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