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Degrees of irreducible characters of the symmetric group and exponential growth


Authors: Antonio Giambruno and Sergey Mishchenko
Journal: Proc. Amer. Math. Soc. 144 (2016), 943-953
MSC (2010): Primary 20C30, 05A17
DOI: https://doi.org/10.1090/proc/12758
Published electronically: October 6, 2015
MathSciNet review: 3447648
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider sequences of degrees of ordinary irreducible $ S_n$-
characters. We assume that the corresponding Young diagrams have rows and columns bounded by some linear function of $ n$ with leading coefficient less than one. We show that any such sequence has at least exponential growth and we compute an explicit bound.


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  • [1] A. Berele and A. Regev, Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras, Adv. in Math. 64 (1987), no. 2, 118-175. MR 884183 (88i:20006), https://doi.org/10.1016/0001-8708(87)90007-7
  • [2] A. Giambruno, S. Mishchenko, and M. Zaicev, Algebras with intermediate growth of the codimensions, Adv. in Appl. Math. 37 (2006), no. 3, 360-377. MR 2261178 (2007g:16033), https://doi.org/10.1016/j.aam.2005.02.005
  • [3] A. Giambruno, S. Mishchenko, and M. Zaicev, Codimensions of algebras and growth functions, Adv. Math. 217 (2008), no. 3, 1027-1052. MR 2383893 (2008m:17005), https://doi.org/10.1016/j.aim.2007.07.008
  • [4] A. Giambruno and M. Zaicev, Exponential codimension growth of PI algebras: an exact estimate, Adv. Math. 142 (1999), no. 2, 221-243. MR 1680198 (2000a:16048), https://doi.org/10.1006/aima.1998.1790
  • [5] Antonio Giambruno and Mikhail Zaicev, Polynomial identities and asymptotic methods, Mathematical Surveys and Monographs, vol. 122, American Mathematical Society, Providence, RI, 2005. MR 2176105 (2006g:16054)
  • [6] Antonio Giambruno and Mikhail Zaicev, Proper identities, Lie identities and exponential codimension growth, J. Algebra 320 (2008), no. 5, 1933-1962. MR 2437638 (2009g:16032), https://doi.org/10.1016/j.jalgebra.2008.06.009
  • [7] Gordon James and Adalbert Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, vol. 16, Addison-Wesley Publishing Co., Reading, Mass., 1981. With a foreword by P. M. Cohn; with an introduction by Gilbert de B. Robinson. MR 644144 (83k:20003)
  • [8] Olga Malyusheva, Sergey Mishchenko, and Andrey Verevkin, Series of varieties of Lie algebras of different fractional exponents, C. R. Acad. Bulgare Sci. 66 (2013), no. 3, 321-330. MR 3114554, https://doi.org/10.7546/CR-2013-66-3-13101331-2
  • [9] S. P. Mishchenko, Lower bounds on the dimensions of irreducible representations of symmetric groups and of the exponents of the exponential of varieties of Lie algebras, Mat. Sb. 187 (1996), no. 1, 83-94 (Russian, with Russian summary); English transl., Sb. Math. 187 (1996), no. 1, 81-92. MR 1380205 (97d:17003), https://doi.org/10.1070/SM1996v187n01ABEH000101
  • [10] S. P. Mishchenko, A. Regev, and M. V. Zaicev, A characterization of P.I. algebras with bounded multiplicities of the cocharacters, J. Algebra 219 (1999), no. 1, 356-368. MR 1707676 (2000e:16026), https://doi.org/10.1006/jabr.1998.7916
  • [11] Richard Rasala, On the minimal degrees of characters of $ S_{n}$, J. Algebra 45 (1977), no. 1, 132-181. MR 0427445 (55 #477)
  • [12] Amitai Regev, Asymptotic values for degrees associated with strips of Young diagrams, Adv. in Math. 41 (1981), no. 2, 115-136. MR 625890 (82h:20015), https://doi.org/10.1016/0001-8708(81)90012-8
  • [13] Amitai Regev, Maximal degrees for Young diagrams in the $ (k,l)$ hook, European J. Combin. 19 (1998), no. 6, 721-726. MR 1642718 (99k:05160), https://doi.org/10.1006/eujc.1997.0200
  • [14] Herbert Robbins, A remark on Stirling's formula, Amer. Math. Monthly 62 (1955), 26-29. MR 0069328 (16,1020e)

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Additional Information

Antonio Giambruno
Affiliation: Dipartimento di Matematica e Informatica, Università di Palermo, Via Archirafi 34, 90123 Palermo, Italy
Email: antonio.giambruno@unipa.it

Sergey Mishchenko
Affiliation: Department of Algebra and Geometric Computations, Ulyanovsk State University, Ulyanovsk 432017, Russia
Email: mishchenkosp@mail.ru

DOI: https://doi.org/10.1090/proc/12758
Keywords: Symmetric group, character, exponential growth
Received by editor(s): June 6, 2014
Received by editor(s) in revised form: February 5, 2015
Published electronically: October 6, 2015
Communicated by: Patricia Hersh
Article copyright: © Copyright 2015 American Mathematical Society

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