For stable sequences D(λ,τ) of SU(N) representations, dimensions satisfy dim(D(λ,τ),N) = (-1)^{Area(λ)+Area(τ)} dim(D(τ,λ),-N) and second-order Casimir eigenvalues satisfy C(D(λ,τ),N) = -C(D(λ^T,τ^T),-N).
On Universal Eigenvalues of Casimir Operator
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Motivated by the universal knot polynomials in the gauge Chern-Simons theory, we show that the values of the second Casimir operator on an arbitrary power of Cartan product of $X_2$ and adjoint representations of simple Lie algebras can be represented in a universal form. We show that it complies with $N\longrightarrow -N$ duality of the same operator for $SO(2n)$ and $Sp(2n)$ algebras (the part of $N\leftrightarrow-N$ duality of gauge $SO(2n)$ and $Sp(2n)$ theories). We discuss the phenomena of non-zero universal values of Casimir operator on zero representations.
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$N \leftrightarrow -N$ duality of SU(N) for stable sequences of representations
For stable sequences D(λ,τ) of SU(N) representations, dimensions satisfy dim(D(λ,τ),N) = (-1)^{Area(λ)+Area(τ)} dim(D(τ,λ),-N) and second-order Casimir eigenvalues satisfy C(D(λ,τ),N) = -C(D(λ^T,τ^T),-N).