E8 Lattice - Lattice Points

Lattice Points

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

• 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 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

• 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 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.