Adjoints in symmetric squares of Lie algebra representations
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866917661989928960 |
|---|---|
| author | Floch, Bruno Le Smilga, Ilia |
| author_facet | Floch, Bruno Le Smilga, Ilia |
| contents | *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 $\mathrm{adj}(G)\otimesΛ^2 V$, 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 $Λ^2 V$. 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 $\mathrm{adj}(G)\otimesΛ^2 V$ for a symplectic (resp. orthogonal) representation $V$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_08489 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Adjoints in symmetric squares of Lie algebra representations Floch, Bruno Le Smilga, Ilia Representation Theory Group Theory 17B10, 17B20, 22E46, 22E47 *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 $\mathrm{adj}(G)\otimesΛ^2 V$, 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 $Λ^2 V$. 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 $\mathrm{adj}(G)\otimesΛ^2 V$ for a symplectic (resp. orthogonal) representation $V$. |
| title | Adjoints in symmetric squares of Lie algebra representations |
| topic | Representation Theory Group Theory 17B10, 17B20, 22E46, 22E47 |
| url | https://arxiv.org/abs/2401.08489 |