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Teor. Veroyatnost. i Primenen., 2006, Volume 51, Issue 1, Pages 47–63 (Mi tvp145)  

This article is cited in 3 scientific papers (total in 4 papers)

Markov measures on Young tableaux and induced representations of an 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 show that the class of so-called Markov representations of the infinite symmetric group $\mathfrak{S}_{N}$, associated with Markov measures on the space of infinite Young tableaux, coincides with the class of simple representations, i.e., inductive limits of representations with simple spectrum. The spectral measure of an arbitrary representation of $\mathfrak{S}_{N}$ with simple spectrum is equivalent to a multi-Markov measure on the space of Young tableaux. We also show that the representations of $\mathfrak{S}_{N}$ induced from the identity representations of two-block Young subgroups are Markov and find explicit formulas for the transition probabilities of the corresponding Markov measures. The induced representations are studied with the help of the tensor model of two-row representations of the symmetric groups; in particular, we deduce explicit formulas for the Gelfand–Tsetlin basis in the tensor models.

Keywords: Markov measures, Young tableaux, induced representations, simple spectrum.

DOI: https://doi.org/10.4213/tvp145

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English version:
Theory of Probability and its Applications, 2007, 51:1, 211–223

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Received: 23.11.2005

Citation: A. M. Vershik, N. V. Tsilevich, “Markov measures on Young tableaux and induced representations of an infinite symmetric group”, Teor. Veroyatnost. i Primenen., 51:1 (2006), 47–63; Theory Probab. Appl., 51:1 (2007), 211–223

Citation in format AMSBIB
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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. A. M. Vershik, “A new approach to the representation theory of the symmetric groups. III. Induced representations and the Frobenius–Young correspondence”, Mosc. Math. J., 6:3 (2006), 567–585  mathnet  crossref  mathscinet  zmath
    2. N. I. Nessonov, “Representations of $\mathfrak{S}_\infty$ admissible with respect to Young subgroups”, Sb. Math., 203:3 (2012), 424–458  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    3. Brightwell G., Luczak M., “Order-Invariant Measures on Fixed Causal Sets”, Comb. Probab. Comput., 21:3 (2012), 330–357  crossref  mathscinet  zmath  isi  elib  scopus
    4. 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
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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