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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