Gespeichert in:
| 1. Verfasser: | |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2312.01839 |
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Inhaltsangabe:
- Schur--Weyl--Jones duality establishes the connection between the commuting actions of the symmetric group $S_{n}$ and the partition algebra $P_{k}(n)$ on the tensor space $\left(\mathbb{C}^n\right)^{\otimes k}.$ We give a refinement of this, determining a subspace of $\left(\mathbb{C}^n\right)^{\otimes k}$ on which we have a version of Schur--Weyl duality for the symmetric groups $S_{n}$ and $S_{k}.$ We use this refinement to construct subspaces of $\left(\mathbb{C}^n\right)^{\otimes k}$ that are isomorphic to certain irreducible representations of $S_{n}\times S_{k}.$ We then use the Weingarten calculus for the symmetric group to obtain an explicit formula for the orthogonal projection from $\left(\mathbb{C}^n\right)^{\otimes k}$ to each subspace.