The article presents the results of numerical computations of statistics related to Young diagrams, including estimates on the maximum and average (with respect to the Plancherel distribution) dimension of irreducible representations of the symmetric group S n . The computed limit shapes of two-dimensional and three-dimensional diagrams distributed according to the Richardson statistics are also presented. Bibliography: 14 titles.
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References
W. Fulton, Young Tableaux, with Applications to Representation Theory and Geometry, Cambridge Univ. Press, Cambridge (1997).
A. M. Vershik and A. Yu. Okounkov, “An inductive method of presenting the theory of representations of symmetric groups,” Russian Math. Surveys, 51, No. 2, 355–356 (1996).
A. M. Vershik and S. V. Kerov, “Asymptotics of the Plancherel measure of the symmetric group and the limiting form of Young tableaux,” Sov. Math. Dokl., 18, 527–531 (1977).
A. M. Vershik and S. V. Kerov, “Asymptotics of maximal and typical dimensions of irreducible representations of asymmetric group,” Funct. Anal. Appl., 19, 21–31 (1985).
H. Rost, “Nonequilibrium behaviour of a many particle process: density profile and local equilibria,” Z. Wahrsch. Verw. Gebiete, 58, No. 1, 41–53 (1981).
A. M. Vershik, A. B. Gribov, and S. V. Kerov, “Experiments in calculating the dimension of a typical representation of the symmetric group,” J. Sov. Math., 28, 568–570 (1983).
D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, Massachusetts (1998).
A. M. Vershik and N. V. Tsilevich, “Induced representations of the infinite symmetric group,” Pure Appl. Math. Quart., 3, No. 4, 1005–1026 (2007).
R. M. Baer and P. Brock, “Natural sorting over permutation spaces,” Math. Comp., 22, 385–410 (1968).
J. McKay, “The largest degrees of irreducible characters of the symmetric group,” Math. Comp., 30, No. 135, 624–631 (1976).
D. Richardson, “Random growth in a tessellation,” Proc. Cambridge Philos. Soc., 74, 515–528 (1973).
A. Vershik and Yu. Yakubovich, “Fluctuation of maximal particle energy of quantum ideal gas and random partitions,” Comm. Math. Phys., 261, No. 3, 759–769 (2006).
R. Cerf and R. Kenyon, “The low-temperature expansion of the Wulff crystal in the 3D Ising model,” Comm. Math. Phys., 222, No. 1, 147–179 (2001).
A. Okounkov and N. Reshetikhin, “Correlation function of Schur process with application to local geometry of a random three-dimensional Young diagram,” J. Amer. Math. Soc., 16, 581–603 (2003).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 373, 2009, pp. 77–93.
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Vershik, A., Pavlov, D. Numerical experiments in problems of asymptotic representation theory. J Math Sci 168, 351–361 (2010). https://doi.org/10.1007/s10958-010-9986-x
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DOI: https://doi.org/10.1007/s10958-010-9986-x