Algebra of double cosets of a symmetric group by a smaller symmetric group
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911115720523776 |
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| author | Neretin, Yury A. |
| author_facet | Neretin, Yury A. |
| contents | Fix a natural $α$. Let $n\ge α$ be an integer. Consider the symmetric group $S_{α+n}$ and its subgroup $S_n$. We consider the group algebra of $S_{α+n}$ and its subalgebra $\mathbb{O}[α;n]$ consisting of $S_n$-biinvariant functions, i.e., functions, which are constant on double cosets of $S_{α+n}$ with respect to $S_n$. We obtain two simple descriptions of $\mathbb{O}[α;n]$. First, we write explicitly formulas for multiplication in a natural basis (structure constants are Pochhammer symbols). Secondly, we describe this algebra in terms of generators and relations. We also construct an interpolating family of algebras $\mathbb{O}[α;ν]$ depending on a complex parameter $ν$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_17069 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Algebra of double cosets of a symmetric group by a smaller symmetric group Neretin, Yury A. Representation Theory Group Theory Rings and Algebras 20C30, 20B30, 20C05, 20N20 Fix a natural $α$. Let $n\ge α$ be an integer. Consider the symmetric group $S_{α+n}$ and its subgroup $S_n$. We consider the group algebra of $S_{α+n}$ and its subalgebra $\mathbb{O}[α;n]$ consisting of $S_n$-biinvariant functions, i.e., functions, which are constant on double cosets of $S_{α+n}$ with respect to $S_n$. We obtain two simple descriptions of $\mathbb{O}[α;n]$. First, we write explicitly formulas for multiplication in a natural basis (structure constants are Pochhammer symbols). Secondly, we describe this algebra in terms of generators and relations. We also construct an interpolating family of algebras $\mathbb{O}[α;ν]$ depending on a complex parameter $ν$. |
| title | Algebra of double cosets of a symmetric group by a smaller symmetric group |
| topic | Representation Theory Group Theory Rings and Algebras 20C30, 20B30, 20C05, 20N20 |
| url | https://arxiv.org/abs/2506.17069 |