E₈ (mathematics) - Subgroups

Subgroups

The smaller exceptional groups E₇ and E₆ sit inside E₈. In the compact group, both E₇×SU(2)/(−1,−1) and E₆×SU(3)/(Z/3Z) are maximal subgroups of E₈.

The 248-dimensional adjoint representation of E₈ may be considered in terms of its restricted representation to the first of these subgroups. It transforms under E₇×SU(2) as a sum of tensor product representations, which may be labelled as a pair of dimensions as (3,1) + (1,133) + (2,56) (since there is a quotient in the product, these notations may strictly be taken as indicating the infinitesimal (Lie algebra) representations). Since the adjoint representation can be described by the roots together with the generators in the Cartan subalgebra, we may see that decomposition by looking at these. In this description:

• (3,1) consists of the roots (0,0,0,0,0,0,1,−1), (0,0,0,0,0,0,−1,1) and the Cartan generator corresponding to the last dimension.
• (1,133) consists of all roots with (1,1), (−1,−1), (0,0), (−½,−½) or (½,½) in the last two dimensions, together with the Cartan generators corresponding to the first 7 dimensions.
• (2,56) consists of all roots with permutations of (1,0), (−1,0) or (½,−½) in the last two dimensions.

The 248-dimensional adjoint representation of E₈, when similarly restricted, transforms under E₆×SU(3) as: (8,1) + (1,78) + (3,27) + (3,27). We may again see the decomposition by looking at the roots together with the generators in the Cartan subalgebra. In this description:

• (8,1) consists of the roots with permutations of (1,−1,0) in the last three dimensions, together with the Cartan generator corresponding to the last two dimensions.
• (1,78) consists of all roots with (0,0,0), (−½,−½,−½) or (½,½,½) in the last three dimensions, together with the Cartan generators corresponding to the first 6 dimensions.
• (3,27) consists of all roots with permutations of (1,0,0), (1,1,0) or (−½,½,½) in the last three dimensions.
• (3,27) consists of all roots with permutations of (−1,0,0), (−1,−1,0) or (½,−½,−½) in the last three dimensions.

The finite quasisimple groups that can embed in (the compact form of) E₈ were found by Griess & Ryba (1999).

The Dempwolff group is a subgroup of (the compact form of) E₈. It is contained in the Thompson sporadic group, which acts on the underlying vector space of the Lie group E₈ but does not preserve the Lie bracket. The Thompson group fixes a lattice and does preserve the Lie bracket of this lattice mod 3, giving an embedding of the Thompson group into E₈(F₃).