Branching rules and Clebsch–Gordan series for E8

Abstract

Branching rules for the second lowest dimensional representation of the exceptional simple Lie algebra of type E8 are given with repsect to all its 14 maximal semisimple subalgebras. This representation of E8 is of dimension 3875, the maximal subalgebras are of types A8, D8, A1-A7, A1-E7, A2-E6, A3-D5, A4-A4, A1-A2-A5, G2-F4, A1-A2, C2, and three nonconjugate subalgebras all of type A1. The Clebsch–Gordan series, necessary for decomposition of the direct product of three representations of dimension 248, are given.

Aspects of the Exceptional Group E8

“…The high dimension (248) of the adjoint irrep of E8 further complicates the problem of resolving Kronecker products and branchings. As a consequence there is a paucity of known results for E8 (McKay et al 1976b; Wybourne and Bowick 1977). In this paper we are able to find quite simple and efficient methods for resolving all Kronecker products of E8 irreps up to the fourth power in the adjoint irrep and thence to resolve the symmetrized third and fourth powers of the adjoint irrep.…”