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On Algebras of Double Cosets of Symmetric Groups with Respect to Young Subgroups
Yu. A. Neretinabc a University of Vienna
b Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow
c Lomonosov Moscow State University
Abstract:
In the group algebra of the symmetric group $G=S_{n_1+\dots+n_\nu}$, we consider the subalgebra $\Delta$ consisting of all functions invariant with respect to left and right shifts by elements of the Young subgroup $H:=S_{n_1}\times \dots \times S_{n_\nu}$. We discuss structure constants of the algebra $\Delta$ and construct an algebra with continuous parameters $n_1,\dots,n_j$ extrapolating algebras $\Delta$, this can also be rewritten as an asymptotic algebra as $n_j\to\infty$ (for fixed $\nu$). We show that there is a natural map from the Lie algebra of the group of pure braids to $\Delta$ (and therefore this Lie algebra acts in spaces of multiplicities of the quasiregular representation of the group $G$ in functions on $G/H$).
Keywords:
symmetric group, double cosets, Lie algebra of the group of braids,
hypergeometric functions, Poisson algebra.
Received: 22.03.2023 Revised: 23.04.2023
Citation:
Yu. A. Neretin, “On Algebras of Double Cosets of Symmetric Groups with Respect to Young Subgroups”, Mat. Zametki, 114:4 (2023), 591–601; Math. Notes, 114:4 (2023), 583–592
Linking options:
https://www.mathnet.ru/eng/mzm13957https://doi.org/10.4213/mzm13957 https://www.mathnet.ru/eng/mzm/v114/i4/p591
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Abstract page: | 157 | Full-text PDF : | 18 | Russian version HTML: | 82 | References: | 25 | First page: | 9 |
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