User:Fropuff/Drafts/Root lattice

Official page: Root lattice

In mathematics, a root lattice is a lattice generated by the elements of a root system. Root lattices provide important examples of lattices, and are important building blocks of more complicated lattices.

The rank of a root lattice is equal to the rank of the associated root system. Any basis for the root system (i.e. a set of simple roots) acts as a basis for the associated lattice. That is, every lattice point can be written as a linear combination of simple roots with integer coefficients.

If Φ is a rank n root system in R^m we shall denote the associated root lattice by Λ(Φ).

Classification

The classification of root lattices follows from the classification of irreducible root systems.

Properties

We list some properties of the irreducible root lattices together with the cubic lattice Zⁿ. Each of the irreducible root lattices are normalized so that the minimal vectors have length squared equal to 2. They are then all even integral lattices. The root systems are just the set of minimal vectors. The cubic lattice is normalized to be an odd integral lattice with minimal vectors of length 1.

In the following table det(Λ) is the determinant of the Gram matrix (or the squared volume of the fundamental parallelotope), h denotes the Coxeter number, and δ denotes the central sphere packing density which is the packing density divided by the volume of a unit n-ball ($V_{n}=\pi ^{n/2}/(n/2)!$).

$\Lambda$	$|\Phi |$	$\det(\Lambda )$	$|{\mbox{Aut}}(\Lambda )|$	$h$	$\delta$
Zn		1	2n n!		$2^{-n}\,$
An	n(n+1)	n+1	2(n+1)!	n+1	${\frac {2^{-n/2}}{\sqrt {n+1}}}$
Dn	2n(n−1)	4	2n n! 1152 (n=4)	2n−2	$2^{-(n+2)/2}\,$
E6	72	3	144·6!	12	${\frac {1}{8{\sqrt {3}}}}$
E7	126	2	8·9!	18	$1 \over 16$
E8	240	1	192·10!	30	$1 \over 16$

References

• Conway, John H. (1999). Sphere Packings, Lattices and Groups (3rd ed.). New York: Springer-Verlag. ISBN 0-387-98585-9. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)