Abstract
We study the asymptotics of sums of powers of normalized dimensions of complex irreducible representations of the symmetric group
as N→∞. We calculate the limit Gibbs measure on the space of Young diagrams. The problem is connected with a one-dimensional model of statistical physics. In the appendix, written by A. M. Pass, numerical and graphical data about the free energy corresponding to the statistical studied are given.
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Literature cited
A. M. Vershik and S. V. Kerov, “Asymptotics of maximal and typical dimensions of irreducible representations of the symmetric group,” Funkts. Anal. Prilozhen.,19, No. 1, 25–36 (1981).
A. M. Vershik and S. V. Kerov, “Asymptotics of the Plancherel measure of the symmetric group and the limit form of Young tableaux,” Dokl. Akad. Nauk SSSR,233, No. 6, 1024–1027 (1977).
T. M. Liggett, “Ergodic theorem for the asymmetric exclusion process,” Trans. Am. Math. Soc.,213, 237–261 (1975).
H. Rost, “Nonequilibrium of a many particle process: Density profile and local equilibria,” Zs. Wahr.,58, 41–53 (1981).
M. Szalay and P. Turan, “On some problems of the statistical theory of partitions with application to the characters of the symmetric group I,” Acta Math. Acad. Sci. Hung.,29, No. 3–4, 361–379 (1977); III,32, No. 1–2, 129–155 (1978).
J. McKay, “The largest degree of irreducible characters of the symmetric group,” Math. Comp.,32, 624–631 (1978).
A. M. Vershik, “Statistics on the partitions of the natural,” in: Fourth International Conference of Probability Theory and Mathematical Stat. Vilnius, 1985, VNU Science Press (1987).
A. M. Vershik “Bijective proof of the Jacobi identity and reconstruction of Young diagrams,” J. Sov. Math.,41, No. 2 (1988).
G. James, Theory of Representations of Symmetric Groups [Russian translation], Moscow (1982).
Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 164, pp. 20–29, 1987.
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Vershik, A.M. A statistical sum associated with Young diagrams. J Math Sci 47, 2379–2386 (1989). https://doi.org/10.1007/BF01840420
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DOI: https://doi.org/10.1007/BF01840420