Vershik, A. M.; Kerov, S. V. Asymptotics of maximal and typical dimensions of irreducible representations of a symmetric group. (English. Russian original) Zbl 0592.20015 Funct. Anal. Appl. 19, 21-31 (1985); translation from Funkts. Anal. Prilozh. 19, No. 1, 25-36 (1985). We give order-sharp two-sided bounds of the largest and the typical dimensions of irreducible representations of the symmetric group \({\mathfrak S}_ N\) for \(N\to \infty\). Both problems are solved simultaneously and are connected with the earlier-proved theorem about the limit form of a typical Young diagram. Some applications of the obtained results are indicated. Cited in 6 ReviewsCited in 68 Documents MSC: 20C30 Representations of finite symmetric groups 05A15 Exact enumeration problems, generating functions Keywords:dimensions; irreducible representations of symmetric groups; Young diagrams PDFBibTeX XMLCite \textit{A. M. Vershik} and \textit{S. V. Kerov}, Funct. Anal. Appl. 19, 21--31 (1985; Zbl 0592.20015); translation from Funkts. Anal. Prilozh. 19, No. 1, 25--36 (1985) Full Text: DOI References: [1] R. M. Baer and P. Brock, ”Natural sorting over permutation spaces,” Math. Comp.,22, 385-410 (1968). [2] R. Rasala, ”On minimal degrees of characters of Sn,” J. Algebra,45, 132-181 (1977). · Zbl 0348.20009 [3] J. McKay, ”The largest degrees of irreducible characters of the symmetric group,” Math. Comp.,32, 624-631 (1978). · Zbl 0345.20011 [4] A. M. Vershik and S. V. Kerov, ”The asymptotic of the Plancherel measure of a symmetric group and the limit form of Young tableaux,” Dokl. Akad. Nauk SSSR,233, No. 6, 1024-1027 (1977). · Zbl 0406.05008 [5] A. M. Vershik, A. B. Gribov, and S. V. Kerov, ”Experiments on the computation of the dimension of a typical representation of a symmetric group,” J. Sov. Math.,28, No. 4 (1985). · Zbl 0557.20006 [6] B. F. Logan and L. A. Shepp, ”A variational problem for random Young tableaux,” Adv. Math.,26, 206-222 (1977). · Zbl 0363.62068 [7] A. B. Gribov, ”The limit form with respect to the Plancherel measure of a Young tableau,” Vestn. Leningr. Gos. Univ., Ser. Mat., No. 1 (1985). [8] D. Knuth, The Art of Computer Programming, Vol. 3, Sorting and Searching, Addison Wesley, Reading (1973). · Zbl 0302.68010 [9] L. Hörmander, Linear Partial Differential Operators, Springer-Verlag, Berlin?New York (1963). · Zbl 0108.09301 [10] A. M. Vershik and S. V. Kerov, ”The asymptotic theory of characters of a symmetric group,” Funkts. Anal. Prilozhen.,15, No. 4, 15-27 (1981). · Zbl 0534.20008 [11] A. M. Vershik, Editor’s Appendix to the Book: G. D. James, The Representation Theory of the Symmetric Groups [Russian translation], Mir, Moscow (1982), pp. 191-213. [12] S. V. Kerov and A. M. Vershik, ”The characters of the infinite symmetric groups and probability properties of the Robinson?Shensted?Knuth algorithm,” SIAM J. Algebra Discr. Math., No. 5 (1984). · Zbl 0545.22001 [13] G. D. James, The Representation Theory of the Symmetric Groups, Lecture Notes in Math., Vol. 682, Springer-Verlag, Berlin?New York (1978). · Zbl 0393.20009 [14] J. F. C. Kingman, ”Subadditive ergodic theorem,” Ann. Prob.,1, No. 6, 883-909 (1973). · Zbl 0311.60018 [15] D. E. Knuth, ”Permutations, matrices and generalized Young tableaux,” Pac. J. Math.,34, No. 3, 709-727 (1970). · Zbl 0199.31901 [16] C. Green and D. J. Kleitman, ”The structure of Sperner’s K-families,” J. Compt. Th., A,20, 41-68 (1976). · Zbl 0363.05006 [17] S. V. Fomin, ”Finite partially ordered sets and Young diagrams,” Dokl. Akad. Nauk SSSR,243, No. 5, 1144-1147 (1978). [18] M. Szalay and P. Turan, ”On some problems of the statistical theory of partitions with application to characters of the symmetric group. I,” Acta Math., Acad. Sci. Hung.,29, Nos. 3-4, 361-379 (1977). · Zbl 0371.10033 [19] A. M. Vershik and A. A. Shmidt, ”Limit measures that arise in the asymptotic theory of symmetric groups,” Teor. Veroyatn. Primen., No. 1, 72-88 (1977); No. 1, 42-54 (1978). · Zbl 0375.60007 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.