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.
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,
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,
The lattices Γ and Γ′ 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 is sometimes called the even coordinate system for E while the lattice Γ′ is called the odd coordinate system. Unless we specify otherwise we shall work in the even coordinate system.
Properties
The E lattice Γ can be characterized as the unique lattice in R with the following properties: It is integral, meaning that all scalar products of lattice elements are integers. It is unimodular, meaning that it is integral, and can be generated by the columns of an 8×8 matrix with determinant ±1 (i.e. the volume of the fundamental parallelotope of the lattice is 1). Equivalently, Γ 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: Γ ⊕ Γ and Γ (constructed in an analogous fashion to Γ. In dimension 24 there are 24 such lattices, called Niemeier lattices. The most important of these is the Leech lattice.
One possible basis for Γ is given by the columns of the (upper triangular) matrix
Γ 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 Γ have length equal to √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 E. The lattice Γ is equal to the E 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 Γ.
Symmetry group
The automorphism group (or symmetry group) of a lattice in R is defined as the subgroup of the orthogonal group O(n) that preserves the lattice. The symmetry group of the E lattice is the Weyl/Coxeter group of type E. This is the group generated by reflections in the hyperplanes orthogonal to the 240 roots of the lattice. Its order is given by
The E Weyl group contains a subgroup of order 128·8! consisting of all permutations of the coordinates and all even sign changes. This subgroup is the Weyl group of type D. The full E Weyl group is generated by this subgroup and the block diagonal matrix H⊕H where H is the Hadamard matrix
Geometry
See 5 honeycomb
The E lattice points are the vertices of the 5 honeycomb, which is composed of regular 8-simplex and 8-orthoplex facets. This honeycomb was first studied by Gosset who called it a 9-ic semi-regular figure (Gosset regarded honeycombs in n dimensions as degenerate n+1 polytopes). In Coxeter's notation, Gosset's honeycomb is denoted by 5 and has the Coxeter-Dynkin diagram: