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arXiv · 2506.13062

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

R. L. Mkrtchyan
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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.

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R. L. Mkrtchyan. 2025-06-16. The Casimir eigenvalues on $ad^{\otimes k}$ of SU(N) are linear on N. https://arxiv.org/abs/2506.13062

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