The Casimir eigenvalues on $a{d}^{\otimes k}$ of SU(N) are linear on N

We consider eigenvalues of the Casimir operator on the naturally defined $\mathit{\text{stable}}\mathit{\text{sequences}}$ of representations of $su(N)$ algebra and prove that eigenvalues are linear over $N$ iff ${\lambda }_1+2{\lambda }_2+\mathrm{.}\mathrm{.}\mathrm{.}+k{\lambda }_k={\lambda }_{N-1}+2{\lambda }_{N-2}+\mathrm{.}\mathrm{.}\mathrm{.}+k{\lambda }_{N-k}$, where ${\lambda }_i$ are Dynkin labels, and ${\lambda }_i=0$ for $k<i<N-k$, with fixed $k$. These representations are exactly those which appear in the decomposition of $ad(su(N){)}^{\otimes k}$, therefore this linearity admits the presentation of eigenvalues in the universal, in Vogel’s sense, form, and supports the hypothesis of universal decomposition of $a{d}^{\otimes k}$ into Casimir eigenspaces.