Abstract:
In the present paper, using the split Casimir operators,
we find the decomposition of the antisymmetric part of
the fifth power of the adjoint representation
$\mathfrak{ad}^{\otimes 5}$.
This decomposition contains, in addition to the representations that appeares
in the decomposition of $\mathfrak{ad}^{\otimes 4}$, only one new
representation of $X_5$. The universal dimension of this representation
for exceptional Lie algebras was proposed in [1].
Our decomposition holds for all Lie algebras.
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