Abstract
We consider eigenvalues of the Casimir operator on the naturally defined stable sequences of representations of \(su(N)\) algebra and prove that eigenvalues are linear over N iff \({{\lambda }_{1}} + 2{{\lambda }_{2}} + ... + k{{\lambda }_{k}} = \) \({{\lambda }_{{N - 1}}} + 2{{\lambda }_{{N - 2}}} + ... + k{{\lambda }_{{N - k}}}\) , 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.
