E8 lattice

In mathematics, the E₈ lattice is a special lattice in R⁸. 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 E₈ root system.

The norm of the E₈ 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 A. Korkin and G. Zolotarev in 1873. The E₈ 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 E₈ lattice is a discrete subgroup of R⁸ of full rank (i.e. it spans all of R⁸). It can be given explicitly by the set of points Γ₈ ⊂ R⁸ such that

In symbols,

It is not hard to check that the sum of two lattice points is another lattice point, so that Γ₈ is indeed a subgroup.

An alternative description of the E₈ lattice which is sometimes convenient is the set of all points in Γ′₈ ⊂ R⁸ such that

In symbols,

The lattices Γ₈ and Γ′₈ are isomorphic and one may pass from one to the other by changing the signs of any odd number of coordinates. The lattice Γ₈ is sometimes called the even coordinate system for E₈ while the lattice Γ₈' is called the odd coordinate system. Unless we specify otherwise we shall work in the even coordinate system.

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