A universal dimension formula for complex simple Lie algebras

Let g be a complex simple Lie algebra. Vogel derived a universal decomposition of S2g into (possibly virtual) Casimir eigenspaces, S2g = C⊕Y2 ⊕Y ′ 2 ⊕Y ′′ 2 which turns out to be a decomposition into irreducible modules. If we let 2t denote the Casimir eigenvalue of the adjoint representation (with respect to some invariant quadratic form), these modules respectively have Casimir eigenvalues 4t − 2α, 4t − 2β, 4t − 2γ, which we may take as the definitions of α, β, γ. Vogel showed that t = α+ β+ γ. He then went on to find Casimir eigenspaces Y3, Y ′ 3 , Y ′′ 3 ⊂ S 3g with eigenvalues 6t−6α, 6t−6β, 6t−6γ (which again turn out to be irreducible), and computed their dimensions through difficult diagrammatic computations and the help of Maple [17]: