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Concept
E8 lattice
Summary
In mathematics, the E_8 lattice is a special lattice in R^8. It can be characterized as the unique positive-definite, even, unimodular lattice of rank 8. The name derives from the fact that it is the root lattice of the E_8 root system.
The norm of the E_8 lattice (divided by 2) is a positive definite even unimodular quadratic form in 8 variables, and conversely such a quadratic form can be used to construct a positive-definite, even, unimodular lattice of rank 8.
The existence of such a form was first shown by H. J. S. Smith in 1867, and the first explicit construction of this quadratic form was given by Korkin and Zolotarev in 1873.
The E_8 lattice is also called the Gosset lattice after Thorold Gosset who was one of the first to study the geometry of the lattice itself around 1900.
The E_8 lattice is a discrete subgroup of R^8 of full rank (i.e. it spans all of R^8). It can be given explicitly by the set of points Γ_8 ⊂ R^8 such that
all the coordinates are integers or all the coordinates are half-integers (a mixture of integers and half-integers is not allowed), and
the sum of the eight coordinates is an even integer.
In symbols,
It is not hard to check that the sum of two lattice points is another lattice point, so that Γ_8 is indeed a subgroup.
An alternative description of the E_8 lattice which is sometimes convenient is the set of all points in Γ′_8 ⊂ R^8 such that
all the coordinates are integers and the sum of the coordinates is even, or
all the coordinates are half-integers and the sum of the coordinates is odd.
In symbols,
The lattices Γ_8 and Γ′_8 are isomorphic and one may pass from one to the other by changing the signs of any odd number of half-integer coordinates. The lattice Γ_8 is sometimes called the even coordinate system for E_8 while the lattice Γ′_8 is called the odd coordinate system. Unless we specify otherwise we shall work in the even coordinate system.
The E_8 lattice Γ_8 can be characterized as the unique lattice in R^8 with the following properties:
It is integral, meaning that all scalar products of lattice elements are integers.
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