In mathematics, the E lattice is a special lattice in R. 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 root system.
The norm of the E 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.
<!-- Conway claims this, but I can't find it in the paper.
In 1877 they constructed the corresponding E lattice explicitly as part of a study of sphere packings.
-->
The E 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.
Lattice points
The E lattice is a discrete subgroup of R of full rank (i.e. it spans all of R). It can be given explicitly by the set of points Γ ⊂ R 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,
:<math>\Gamma_8 = \left\{(x_i) \in \mathbb Z^8 \cup (\mathbb Z + \tfrac{1}{2})^8 : {\textstyle\sum_i} x_i \equiv 0\;(\mbox{mod }2)\right\}.</math>
It is not hard to check that the sum of two lattice points is another lattice point, so that Γ is indeed a subgroup.
An alternative description of the E lattice which is sometimes convenient is the set of all points in Γ′ ⊂ R 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,
:<math>\Gamma_8' = \left\{(x_i) \in \mathbb Z^8 \cup (\mathbb Z + \tfrac{1}{2})^8 :