Figure 47: Level decomposition of the adjoint representation of . We have displayed the
decomposition up to positive level . At level zero we have the adjoint representation
of and the singlet representation defined by the simple Cartan
generator . Ascending to level one with the root (green vector) gives the lowest weight
of the representation . The weights of labelled by white crosses are on the
lightcone and so their norm squared is zero. At level two we find the lowest weight (blue vector)
of the 15-dimensional representation . 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 . Since these also have multiplicity 2 as roots of
we find that the outer multiplicity of this representation is one, .
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