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

Abstract

We consider eigenvalues of the Casimir operator on the naturally defined \textit{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 $ad^{\otimes k}$ into Casimir eigenspaces.