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Forum Mathematicum

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Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna


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Volume 29, Issue 3

Issues

Character correspondences for symmetric groups and wreath products

Anton Evseev
  • Corresponding author
  • School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, United Kingdom
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Published Online: 2016-08-12 | DOI: https://doi.org/10.1515/forum-2013-0043

Abstract

The Alperin–McKay conjecture relates irreducible characters of a block of an arbitrary finite group to those of its p-local subgroups. A refinement of this conjecture was stated by the author in a previous paper. We prove that this refinement holds for all blocks of symmetric groups. Along the way we identify a “canonical” isometry between the principal block of Spw and that of SpSw. We also prove a general theorem on expressing virtual characters of wreath products in terms of certain induced characters. Much of the paper generalises character-theoretic results on blocks of symmetric groups with abelian defect and related wreath products to the case of arbitrary defect.

Keywords: character; symmetric group; wreath product; McKay conjecture

MSC 2010: 20C30; 20C15

References

  • [1]

    Alperin J. L., The main problem of block theory, Proceedings of the Conference on Finite Groups (Park City 1975), Academic Press, New York (1976), 341–356. Google Scholar

  • [2]

    Barekat F., Reiner V. and van Willigenburg S., Corrigendum to “Coincidences among skew Schur functions”, Adv. Math. 220 (2009), no. 5, 1655–1656. Google Scholar

  • [3]

    Brauer R., Applications of induced characters, Amer. J. Math. 69 (1947), 709–716. Google Scholar

  • [4]

    Broué M., Isométries parfaites, types de blocs, catégories derivées, Représentations Linéaires des Groupes Finis (Luminy 1988), Astérisque 181–182, Société Mathématique de France, Paris (1990), 61–92. Google Scholar

  • [5]

    Chuang J. and Kessar R., Symmetric groups, wreath products, Morita equivalences, and Broué’s abelian defect group conjecture, Bull. Lond. Math. Soc. 34 (2002), no. 2, 174–184. Google Scholar

  • [6]

    Chuang J. and Rouquier R., Derived equivalences for symmetric groups and 𝔰𝔩2-categorification, Ann. of Math. (2) 167 (2008), 245–298. Google Scholar

  • [7]

    Chuang J. and Tan K. M., Filtrations in Rouquier blocks of symmetric groups and Schur algebras, Proc. Lond. Math. Soc. (3) 86 (2003), no. 3, 685–706. Google Scholar

  • [8]

    Curtis C. W. and Reiner I., Methods of Representation Theory, Vol. 1, John Wiley & Sons, New York, 1981. Google Scholar

  • [9]

    Curtis C. W. and Reiner I., Methods of Representation Theory, Vol. 2, John Wiley & Sons, New York, 1987. Google Scholar

  • [10]

    Dade E. C., Counting characters in blocks. II, J. Reine Angew. Math. 448 (1994), 97–190. Google Scholar

  • [11]

    Enguehard M., Isométries parfaites entre blocs de groupes symétriques, Représentations Linéaires des Groupes Finis (Luminy 1988), Astérisque 181–182, Société Mathématique de France, Paris (1990), 157–171. Google Scholar

  • [12]

    Evseev A., The McKay conjecture and Brauer’s induction theorem, Proc. Lond. Math. Soc. (3) 106 (2013), 1248–1290. Google Scholar

  • [13]

    Farahat H. K., On the representations of the symmetric group, Proc. Lond. Math. Soc. (3) 4 (1954), 303–316. Google Scholar

  • [14]

    Farahat H. K., On Schur functions, Proc. Lond. Math. Soc. (3) 8 (1958), 621–630. Google Scholar

  • [15]

    Fong P., The Isaacs–Navarro conjecture for symmetric groups, J. Algebra 260 (2003), no. 1, 154–161. Google Scholar

  • [16]

    Gramain J.-B., On defect groups for generalized blocks of the symmetric group, J. Lond. Math. Soc. (2) 78 (2008), 155–171. Google Scholar

  • [17]

    Isaacs I. M., Character Theory of Finite Groups, Dover, New York, 1994. Google Scholar

  • [18]

    Isaacs I. M. and Navarro G., New refinements of the McKay conjecture for arbitrary finite groups, Ann. of Math. (2) 156 (2002), 333–344. Google Scholar

  • [19]

    James G. D. and Kerber A., The Representation Theory of the Symmetric Group, Addison-Wesley, Reading, 1981. Google Scholar

  • [20]

    Külshammer B., Olsson J. B. and Robinson G. R., Generalized blocks for symmetric groups, Invent. Math. 151 (2003), no. 3, 513–552. Google Scholar

  • [21]

    Macdonald I. G., On the degrees of the irreducible representations of symmetric groups, Bull. Lond. Math. Soc. 3 (1971), 189–192. Google Scholar

  • [22]

    Macdonald I. G., Symmetric Functions and Hall Polynomials, 2nd ed., Oxford University Press, Oxford, 1995. Google Scholar

  • [23]

    Marcus A., On equivalences between blocks of group algebras: Reduction to the simple components, J. Algebra 184 (1996), no. 2, 372–396. Google Scholar

  • [24]

    Narasaki R. and Uno K., Isometries and extra special Sylow groups of order p3, J. Algebra 322 (2009), no. 6, 2027–2068. Google Scholar

  • [25]

    Navarro G., Characters and Blocks of Finite Groups, Cambridge University Press, Cambridge, 1998. Google Scholar

  • [26]

    Olsson J. B., McKay numbers and heights of characters, Math. Scand. 38 (1976), 25–42. Google Scholar

  • [27]

    Rouquier R., Isométries parfaites dans les blocs à défaut abélien des groupes symétriques et sporadiques, J. Algebra 168 (1994), 648–694. Google Scholar

About the article


Received: 2013-03-19

Revised: 2015-10-28

Published Online: 2016-08-12

Published in Print: 2017-05-01


Funding Source: Engineering and Physical Sciences Research Council

Award identifier / Grant number: EP/G050244

The author was supported by the EPSRC Postdoctoral Fellowship EP/G050244.


Citation Information: Forum Mathematicum, Volume 29, Issue 3, Pages 581–616, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2013-0043.

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