Limits of group algebras for growing symmetric groups and wreath products
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915225447432192 |
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| author | Devyatkova, Irina Olshanski, Grigori |
| author_facet | Devyatkova, Irina Olshanski, Grigori |
| contents | Let $S(\infty)$ denote the infinite symmetric group formed by the finitary permutations of the set of natural numbers; this is a countable group. We introduce its virtual group algebra, a completion of the conventional group algebra $\mathbb C[S(\infty)]$. The virtual group algebra is obtained by taking large-$n$ limits of the finite-dimensional group algebras $\mathbb C[S(n)]$ in the so-called tame representations of $S(\infty)$. We establish a connection with the centralizer construction of Molev-Olshanski [J. Algebra, 237 (2001), 302-341; arXiv:math/0002165] and Drinfeld-Lusztig degenerate affine Hecke algebras. This makes it possible to describe the structure of the virtual group algebra. Then we extend the results to wreath products $G\wr S(\infty)$ with arbitrary finite groups $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_02410 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Limits of group algebras for growing symmetric groups and wreath products Devyatkova, Irina Olshanski, Grigori Representation Theory Rings and Algebras Let $S(\infty)$ denote the infinite symmetric group formed by the finitary permutations of the set of natural numbers; this is a countable group. We introduce its virtual group algebra, a completion of the conventional group algebra $\mathbb C[S(\infty)]$. The virtual group algebra is obtained by taking large-$n$ limits of the finite-dimensional group algebras $\mathbb C[S(n)]$ in the so-called tame representations of $S(\infty)$. We establish a connection with the centralizer construction of Molev-Olshanski [J. Algebra, 237 (2001), 302-341; arXiv:math/0002165] and Drinfeld-Lusztig degenerate affine Hecke algebras. This makes it possible to describe the structure of the virtual group algebra. Then we extend the results to wreath products $G\wr S(\infty)$ with arbitrary finite groups $G$. |
| title | Limits of group algebras for growing symmetric groups and wreath products |
| topic | Representation Theory Rings and Algebras |
| url | https://arxiv.org/abs/2504.02410 |