Vershik, A. M.; Tsilevich, N. V. Markov measures on Young tableaux and induced representations of the infinite symmetric group. (English. Russian original) Zbl 1155.20013 Theory Probab. Appl. 51, No. 1, 211-223 (2007); translation from Teor. Veroyatn. Primen. 51, No. 1, 47-63 (2006). Summary: We show that the class of so-called Markov representations of the infinite symmetric group \(\mathfrak S_{\mathbf N}\), 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 \(\mathfrak S_{\mathbf N}\) with simple spectrum is equivalent to a multi-Markov measure on the space of Young tableaux. We also show that the representations of \(\mathfrak S_{\mathbf N}\) 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. Cited in 2 Documents MSC: 20C32 Representations of infinite symmetric groups 05E10 Combinatorial aspects of representation theory 60B15 Probability measures on groups or semigroups, Fourier transforms, factorization 22D30 Induced representations for locally compact groups Keywords:infinite symmetric group; Markov measures; infinite Young tableaux; induced representations; simple spectra of representations; Gelfand-Tsetlin bases PDFBibTeX XMLCite \textit{A. M. Vershik} and \textit{N. V. Tsilevich}, Theory Probab. Appl. 51, No. 1, 211--223 (2007; Zbl 1155.20013); translation from Teor. Veroyatn. Primen. 51, No. 1, 47--63 (2006) Full Text: DOI arXiv