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

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

Famous quotes containing the words symmetry and/or group:

“What makes a regiment of soldiers a more noble object of view than the same mass of mob? Their arms, their dresses, their banners, and the art and artificial symmetry of their position and movements.”
—George Gordon Noel Byron (1788–1824)

“Jury—A group of twelve men who, having lied to the judge about their hearing, health, and business engagements, have failed to fool him.”
—H.L. (Henry Lewis)