Tensor invariants for classical groups revisited
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866917915672969216 |
|---|---|
| author | Erickson, William Q. Hunziker, Markus |
| author_facet | Erickson, William Q. Hunziker, Markus |
| contents | We reconsider an old problem, namely the dimension of the $G$-invariant subspace in $V^{\otimes p} \otimes V^{*\otimes q}$, where $G$ is one of the classical groups ${\rm GL}(V)$, ${\rm SL}(V)$, ${\rm O}(V)$, ${\rm SO}(V)$, or ${\rm Sp}(V)$. Spanning sets for the invariant subspace have long been well known, but linear bases are more delicate. The main contribution of this paper is a combinatorial realization of linear bases via standard Young tableaux and arc diagrams, in a uniform manner for all five classical groups. As a secondary contribution, we survey the many equivalent ways -- some old, some new -- to enumerate the elements in these bases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_17496 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tensor invariants for classical groups revisited Erickson, William Q. Hunziker, Markus Combinatorics Representation Theory 05E10 (Primary) 16W22, 05A19 (Secondary) We reconsider an old problem, namely the dimension of the $G$-invariant subspace in $V^{\otimes p} \otimes V^{*\otimes q}$, where $G$ is one of the classical groups ${\rm GL}(V)$, ${\rm SL}(V)$, ${\rm O}(V)$, ${\rm SO}(V)$, or ${\rm Sp}(V)$. Spanning sets for the invariant subspace have long been well known, but linear bases are more delicate. The main contribution of this paper is a combinatorial realization of linear bases via standard Young tableaux and arc diagrams, in a uniform manner for all five classical groups. As a secondary contribution, we survey the many equivalent ways -- some old, some new -- to enumerate the elements in these bases. |
| title | Tensor invariants for classical groups revisited |
| topic | Combinatorics Representation Theory 05E10 (Primary) 16W22, 05A19 (Secondary) |
| url | https://arxiv.org/abs/2401.17496 |