Figure 47

Figure 47: Level decomposition of the adjoint representation of AE3. We have displayed the decomposition up to positive level ℓ = 2. At level zero we have the adjoint representation (0) ℛ 1 = 80 of 𝔰𝔩(3,ℝ ) and the singlet representation (0) ℛ 2 = 10 defined by the simple Cartan generator ∨ α 1. Ascending to level one with the root α1 (green vector) gives the lowest weight Λ (1) of the representation ℛ (1) = 61. The weights of ℛ (1) labelled by white crosses are on the lightcone and so their norm squared is zero. At level two we find the lowest weight Λ(2) (blue vector) of the 15-dimensional representation ℛ (2) = 15 2. Again, the white crosses label weights that are on the lightcone. The three innermost weights are inside of the lightcone and the rings indicate that these all have multiplicity 2 as weights of (2) ℛ. Since these also have multiplicity 2 as roots of ⋆ 𝔥 𝔤 we find that the outer multiplicity of this representation is one, μ(ℛ (2)) = 1.