Geodesic
On the Fourier transform on the infinite symmetric group
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XII, Tome 325 (2005), pp. 61-82 Cet article a éte moissonné depuis la source Math-Net.Ru

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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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A. M. Vershik; N. V. Tsilevich. On the Fourier transform on the infinite symmetric group. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XII, Tome 325 (2005), pp. 61-82. http://geodesic.mathdoc.fr/item/ZNSL_2005_325_a3/

[1] G. Dzheims, Teoriya predstavlenii simmetricheskikh grupp, Mir, M., 1982 | MR

[2] G. James, A. Kerber, The Representation Theory of the Symmetric Group, Addison-Wesley, Reading, Mass., 1981 | MR

[3] E. Khyuitt, K. Ross, Abstraktnyi garmonicheskii analiz, T. 1, Nauka, M., 1974; Т. 2, Мир

[4] S. Kerov, G. Olshanski, A. Vershik, “Harmonic analysis on the infinite symmetric group. A deformation of the regular representation.”, C. R. Acad. Sci. Paris Ser. I. Math., 316:8 (1993), 773–778 | MR | Zbl

[5] S. Kerov, G. Olshanski, A. Vershik, “Harmonic analysis on the infinite symmetric group”, Invent. Math., 158:3 (2004), 551–642 | DOI | MR | Zbl

[6] I. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition, Clarendon Press, Oxford, 1995 | MR | Zbl

[7] A. Yu. Okunkov, “Teorema Toma i predstavleniya beskonechnoi bisimmetricheskoi gruppy”, Funkts. anal. i pril., 28:2 (1994), 31–40 | MR

[8] G. I. Olshanskii, Unitarnye predstavleniya $(G, K)$-par svyazannykh s beskonechnoi simmetricheskoi gruppoi $S(\infty)$, Algebra i analiz, 1, no. 4, 1989 | MR

[9] G. K. Pedersen, $C^*$-algebras and Their Automorphism Groups, Academic Press, London–New York, 1979 | MR | Zbl

[10] J. Renault, A Groupoid Approach to C$^*$-Algebras, Lect. Notes in Math., 793, Springer-Verlag, Berlin–Heildelberg–New York, 1980 | MR | Zbl

[11] R. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge Univ. Press, Cambridge, 1999 | MR

[12] A. Vershik, “Gelfand–Tsetlin algebras, expectations, inverse limits, Fourier analysis”, Unity of Mathematics. Proc. of the conference dedicated to I. M. Gelfand, Birkhäuser, 2005 (to appear) | MR

[13] A. M. Vershik, S. V. Kerov, “Asimptoticheskaya teoriya kharakterov simmetricheskoi gruppy”, Funkts. anal. i pril., 15:4 (1981), 15–27 | MR | Zbl

[14] A. M. Vershik, S. V. Kerov, “The Grothendieck group of infinite symmetric group and symmetric functions (with the elements of the theory of K$_0$-functor of AF-algebras)”, Representation of Lie Groups and Related Topics, eds. A. M. Vershik, D. P. Zhelobenko, Gordon and Breach Sci. Publ., 1990, 39–118 | MR

[15] A. M. Vershik, A. Yu. Okunkov, “Novyi podkhod k teorii predstavlenii simmetricheskikh grupp, II”, Zap. nauchn. semin. POMI, 307, 2004, 57–98 | MR

[16] A. M. Vershik, A. Yu. Okunkov, “Novyi podkhod k teorii predstavlenii simmetricheskikh grupp, II”, Prilozhenie k knige U. Fulton, Tablitsy Yunga, MTsNMO, M., 2004