Adjoints in symmetric squares of Lie algebra representations
| Autor: | Floch, Bruno Le, Smilga, Ilia |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | *Caveat: we learned post-factum that most of these results are not novel. We are keeping this paper for continuity reasons.* Given finite-dimensional complex representations $V$ and $V'$ of a simply-connected semisimple compact Lie group $G$, we determine the dimension of the $G$-invariant subspace of $\mathrm{adj}(G)\otimes V\otimes V'$, of $\mathrm{adj}(G)\otimes S^2 V$, and of $\mathrm{adj}(G)\otimes\Lambda^2 V$, where $\mathrm{adj}(G)$ is the adjoint representation. In other words we derive the multiplicity with which summands of $\mathrm{adj}(G)$ appear in a tensor product $V \otimes V'$ or (anti)symmetric square $S^2 V$ or $\Lambda^2 V$. We find in particular that the dimension of the $G$-invariant subspace of $\mathrm{adj}(G)\otimes S^2 V$ is larger than (resp. smaller or equal to) that of $\mathrm{adj}(G)\otimes\Lambda^2 V$ for a symplectic (resp. orthogonal) representation $V$. Comment: 14 pages. The $\mu = \overline{\nu}$ case of our Theorem 1.3, as well as our Theorem 1.4, already appear in the literature; we added the relevant references |
| Databáze: | arXiv |
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