Properties
The E8 lattice Γ8 can be characterized as the unique lattice in R8 with the following properties:
- It is unimodular, meaning that it can be generated by the columns of a 8×8 matrix with determinant ±1 (i.e. the volume of the fundamental parallelotope of the lattice is 1). Equivalently, Γ8 is self-dual, meaning it is equal to its dual lattice.
- It is even, meaning that the norm of any lattice vector is even.
Even unimodular lattices can occur only in dimensions divisible by 8. In dimension 16 there are two such lattices: Γ8 ⊕ Γ8 and Γ16 (constructed in an analogous fashion to Γ8). In dimension 24 there are 24 such lattices, called Niemeier lattices. The most important of these is the Leech lattice.
One possible basis for Γ8 is given by the columns of the (upper triangular) matrix
![\left[\begin{smallmatrix}
2 & -1 & 0 & 0 & 0 & 0 & 0 & 1/2 \\
0 & 1 & -1 & 0 & 0 & 0 & 0 & 1/2 \\
0 & 0 & 1 & -1 & 0 & 0 & 0 & 1/2 \\
0 & 0 & 0 & 1 & -1 & 0 & 0 & 1/2 \\
0 & 0 & 0 & 0 & 1 & -1 & 0 & 1/2 \\
0 & 0 & 0 & 0 & 0 & 1 & -1 & 1/2 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 1/2 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/2
\end{smallmatrix}\right]](http://upload.wikimedia.org/math/f/4/1/f41a5e00ea3e960eb81ce72369f481ba.png)
Γ8 is then the integral span of these vectors. All other possible bases are obtained from this one by right multiplication by elements of GL(8,Z).
The shortest nonzero vectors in Γ8 have norm 2. There are 240 such vectors.
- All half-integer: (can only be ±1/2)
- All positive or all negative: 2
- Four positive, four negative: (8*7*6*5)/(4*3*2*1)=70
- Two of one, six of the other: 2*(8*7)/(2*1) = 56
- All integer: (can only be 0, ±1)
- Two ±1, six zeroes: 4*(8*7)/(2*1)=112
These form a root system of type E8. The lattice Γ8 is equal to the E8 root lattice, meaning that it is given by the integral span of the 240 roots. Any choice of 8 simple roots gives a basis for Γ8.
Read more about this topic: E8 Lattice
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