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| Format: | Preprint |
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2025
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| Accès en ligne: | https://arxiv.org/abs/2506.13062 |
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| _version_ | 1866908556287016960 |
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| author | Mkrtchyan, R. L. |
| author_facet | Mkrtchyan, R. L. |
| contents | 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 $λ_1+2λ_2+...+kλ_k=λ_{N-1}+2λ_{N-2}+...+kλ_{N-k}$, where $λ_i$ are Dynkin labels, and $λ_i=0$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_13062 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Casimir eigenvalues on $ad^{\otimes k}$ of SU(N) are linear on N Mkrtchyan, R. L. Mathematical Physics High Energy Physics - Theory 17B10, 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 $λ_1+2λ_2+...+kλ_k=λ_{N-1}+2λ_{N-2}+...+kλ_{N-k}$, where $λ_i$ are Dynkin labels, and $λ_i=0$ 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. |
| title | The Casimir eigenvalues on $ad^{\otimes k}$ of SU(N) are linear on N |
| topic | Mathematical Physics High Energy Physics - Theory 17B10, |
| url | https://arxiv.org/abs/2506.13062 |