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Zap. Nauchn. Sem. POMI, 2005, Volume 325, Pages 61–82 (Mi znsl350)  

This article is cited in 6 scientific papers (total in 7 papers)

On the Fourier transform on the infinite symmetric group

A. M. Vershik, N. V. Tsilevich

St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences

Abstract: We present a sketch of the Fourier theory on the infinite symmetric group ${\mathfrak S}_\infty$. As a dual space to ${\mathfrak S}_\infty$, we suggest the space (groupoid) of Young bitableaux $\mathcal B$. The Fourier transform of a function on the infinite symmetric group is a martingale with respect to the so-called full Plancherel measure on the groupoid of bitableaux. The Plancherel formula determines an isometry of the space $l^2({\mathfrak S}_\infty,m)$ of square integrable functions on the infinite symmetric group with the counting measure and the space $L^2({\mathcal B},\tilde\mu)$ of square integrable functions on the groupoid of bitableaux with the full Plancherel measure.

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English version:
Journal of Mathematical Sciences (New York), 2006, 138:3, 5663–5673

Bibliographic databases:

UDC: 517.986
Received: 25.05.2005

Citation: A. M. Vershik, N. V. Tsilevich, “On the Fourier transform on the infinite symmetric group”, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XII, Zap. Nauchn. Sem. POMI, 325, POMI, St. Petersburg, 2005, 61–82; J. Math. Sci. (N. Y.), 138:3 (2006), 5663–5673

Citation in format AMSBIB
\Bibitem{VerTsi05}
\by A.~M.~Vershik, N.~V.~Tsilevich
\paper On the Fourier transform on the infinite symmetric group
\inbook Representation theory, dynamical systems, combinatorial and algoritmic methods. Part~XII
\serial Zap. Nauchn. Sem. POMI
\yr 2005
\vol 325
\pages 61--82
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl350}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2160319}
\zmath{https://zbmath.org/?q=an:1078.43002}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2006
\vol 138
\issue 3
\pages 5663--5673
\crossref{https://doi.org/10.1007/s10958-006-0334-0}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33748640593}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. P. P. Nikitin, “The centralizer algebra of the diagonal action of the group $GL_n(\mathbb C)$ in a mixed tensor space”, J. Math. Sci. (N. Y.), 141:4 (2007), 1479–1493  mathnet  crossref  mathscinet  zmath  elib  elib
    2. Tsilevich N.V., Vershik A.M., “On Different Models of Representations of the Infinite Symmetric Group”, Adv. Appl. Math., 37:4 (2006), 526–540  crossref  mathscinet  zmath  isi  elib  scopus
    3. Tsilevich N.V., Vershik A.M., “Induced representations of the infinite symmetric group”, Pure Appl. Math. Q., 3:4, Part 1 (2007), 1005–1026  crossref  mathscinet  zmath  isi  elib  scopus
    4. Vershik A.M. Tsilevich N.V., “Induced Representations of the Infinite Symmetric Group and their Spectral Theory”, Dokl. Math., 75:1 (2007), 1–4  crossref  mathscinet  zmath  isi  elib  scopus
    5. Strahov E., “$Z$-measures on partitions related to the infinite Gelfand pair $(S(2\infty),H(\infty))$”, J. Algebra, 323:2 (2010), 349–370  crossref  mathscinet  zmath  isi  elib  scopus
    6. Oner T., “Infinite Symmetric Groups”, ARS Comb., 107 (2012), 129–140  mathscinet  zmath  isi  elib
    7. V. M. Buchstaber, M. I. Gordin, I. A. Ibragimov, V. A. Kaimanovich, A. A. Kirillov, A. A. Lodkin, S. P. Novikov, A. Yu. Okounkov, G. I. Olshanski, F. V. Petrov, Ya. G. Sinai, L. D. Faddeev, S. V. Fomin, N. V. Tsilevich, Yu. V. Yakubovich, “Anatolii Moiseevich Vershik (on his 80th birthday)”, Russian Math. Surveys, 69:1 (2014), 165–179  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
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