Abstract
In this note, we construct a 2-basic set of the alternating group \({\mathfrak{A}_n}\). To do this, we construct a 2-basic set of the symmetric group \({\mathfrak{S}_n}\) with an additional property, such that its restriction to \({\mathfrak{A}_n}\) is a 2-basic set. We adapt here a method developed by Brunat and Gramain (J. Reine Angew. Math., to appear) for the case when the characteristic is odd. One of the main tools is the generalized perfect isometries defined by Külshammer et al. (Invent. Math. 151, 513–552, (2003)).
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References
C. Bessenrodt, A 2-block splitting in alternating groups, to appear in Algebra Number Theory.
Brunat O.: Basic sets in defining characteristic for general linear groups of small rank. J. Pure Appl. Algebra 213, 698–710 (2009)
O. Brunat and J.-B. Gramain, A basic set for the alternating group, to appear in J. Reine Angew. Math.
Geck M.: Basic sets of Brauer characters of finite groups of Lie type. II. J. London Math. Soc. (2) 47, 255–268 (1993)
Geck M.: Basic sets of Brauer characters of finite groups of Lie type. III. Manuscripta Math. 85, 195–216 (1994)
Geck M., Hiss G.: Basic sets of Brauer characters of finite groups of Lie type. J. Reine Angew. Math. 418, 173–188 (1991)
M. Geck and G. Pfeiffer, Characters of finite Coxeter groups and Iwahori-Hecke algebras, London Mathematical Society Monographs. New Series 21, The Clarendon Press Oxford University Press, New York, 2000.
Gramain J.-B.: On defect groups for generalized blocks of the symmetric group. J. London Math. Soc. 78, 155–171 (2008)
G. James and A. Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications 16, Addison-Wesley Publishing Co., Reading, Mass., 1981.
Külshammer B., Olsson J., Robinson G.R.: Generalized blocks for symmetric groups. Invent. Math. 151, 513–552 (2003)
Osima M.: On the representations of the generalized symmetric group. Math. J. Okayama Univ. 4, 39–56 (1954)
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Brunat, O., Gramain, JB. A 2-basic set of the alternating group. Arch. Math. 94, 301–309 (2010). https://doi.org/10.1007/s00013-010-0116-2
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DOI: https://doi.org/10.1007/s00013-010-0116-2