E8 Lattice - Symmetry Group

Symmetry Group

The automorphism group (or symmetry group) of a lattice in Rn is defined as the subgroup of the orthogonal group O(n) that preserves the lattice. The symmetry group of the E8 lattice is the Weyl/Coxeter group of type E8. This is the group generated by reflections in the hyperplanes orthogonal to the 240 roots of the lattice. Its order is given by

The E8 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 D8. The full E8 Weyl group is generated by this subgroup and the block diagonal matrix H4⊕H4 where H4 is the Hadamard matrix

H_4 = \tfrac{1}{2}\left[\begin{smallmatrix} 1 & 1 & 1 & 1\\ 1 & -1 & 1 & -1\\ 1 & 1 & -1 & -1\\ 1 & -1 & -1 & 1\\ \end{smallmatrix}\right]

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