Projection formulas and a refinement of Schur--Weyl--Jones duality for symmetric groups
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866916724019822592 |
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| author | Cassidy, Ewan |
| author_facet | Cassidy, Ewan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_01839 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Projection formulas and a refinement of Schur--Weyl--Jones duality for symmetric groups Cassidy, Ewan Representation Theory 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. |
| title | Projection formulas and a refinement of Schur--Weyl--Jones duality for symmetric groups |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2312.01839 |