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On different models of representations of the infinite symmetric group. (English) Zbl 1149.20013

Summary: We present an explicit description of the isomorphism between two models of finite factor representations of the infinite symmetric group: the tableau model in the space of functions on Young bitableaux and the dynamical model in the space of functions on pairs of Bernoulli sequences. The main tool used is the Fourier transform on the symmetric groups. We also start the investigation of the so-called tensor model of two-row representations of the symmetric groups, which plays an intermediate role between the tableau and dynamical models, and show its relations to both these models.

MSC:

20C32 Representations of infinite symmetric groups
05E10 Combinatorial aspects of representation theory
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[1] James, G., The Representation Theory of the Symmetric Group (1978), Springer: Springer Berlin
[2] James, G.; Kerber, A., The Representation Theory of the Symmetric Group (1981), Addison-Wesley: Addison-Wesley Reading, MA
[3] Nikitin, P. P., A realization of the irreducible representations of \(S_n\) corresponding to 2-row diagrams in square-free symmetric multilinear forms, Zap. Nauchn. Sem. St.-Petersburg. Otdel. Mat. Inst. Steklov. (POMI). Zap. Nauchn. Sem. St.-Petersburg. Otdel. Mat. Inst. Steklov. (POMI), J. Math. Sci. (New York), 129, 2, 3796-3799 (2005), English translation: · Zbl 1144.20301
[4] Olshanski, G. I., Unitary representations of \((G, K)\)-pairs that are connected with the infinite symmetric group \(S(\infty)\), Leningrad Math. J., 1, 4, 983-1014 (1990) · Zbl 0731.20009
[5] Renault, J., A Groupoid Approach to \(C^*\)-Algebras, Lecture Notes in Math., vol. 793 (1980), Springer-Verlag: Springer-Verlag Berlin-Heidelberg-New York · Zbl 0433.46049
[6] Strǎtilǎ, Ş.; Voiculescu, D., Representations of AF-Algebras and of the Group \(U(\infty)\), Lecture Notes in Math., vol. 486 (1975), Springer-Verlag: Springer-Verlag Berlin-Heidelberg-New York · Zbl 0318.46069
[7] Vershik, A. M.; Kerov, S. V., Characters and factor representations of the infinite symmetric group, Sov. Math. Dokl., 23, 389-392 (1981) · Zbl 0534.20008
[8] Vershik, A. M.; Kerov, S. V., Asymptotic theory of characters of the symmetric group, Funct. Anal. Appl., 15, 246-255 (1982) · Zbl 0507.20006
[9] Vershik, A. M.; Kerov, S. V., The Grothendieck group of infinite symmetric group and symmetric functions (with the elements of the theory of \(K_0\)-functor of AF-algebras), (Vershik, A. M.; Zhelobenko, D. P., Representation of Lie Groups and Related Topics (1990), Gordon and Breach), 39-118 · Zbl 0723.20005
[10] Vershik, A. M.; Tsilevich, N. V., On the Fourier transform on the infinite symmetric group, Zap. Nauchn. Sem. St.-Petersburg. Otdel. Mat. Inst. Steklov. (POMI), 325, 61-82 (2005) · Zbl 1078.43002
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