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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Maximal ideals in modular group algebras
of the finitary symmetric and
alternating groups


Authors: Alexander Baranov and Alexander Kleshchev
Journal: Trans. Amer. Math. Soc. 351 (1999), 595-617
MSC (1991): Primary 20C05, 16S34
MathSciNet review: 1443188
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Abstract: The main result of the paper is a description of the maximal ideals in the modular group algebras of the finitary symmetric and alternating groups (provided the characteristic $p$ of the ground field is greater than 2). For the symmetric group there are exactly $p-1$ such ideals and for the alternating group there are $(p-1)/2$ of them. The description is obtained in terms of the annihilators of certain systems of the `completely splittable' irreducible modular representations of the finite symmetric and alternating groups. The main tools used in the proofs are the modular branching rules (obtained earlier by the second author) and the `Mullineux conjecture' proved recently by Ford-Kleshchev and Bessenrodt-Olsson. The results obtained are relevant to the theory of PI-algebras. They are used in a later paper by the authors and A. E. Zalesskii on almost simple group algebras and asymptotic properties of modular representations of symmetric groups.


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Additional Information

Alexander Baranov
Affiliation: Institute of Mathematics, Academy of Sciences of Belarus, Surganova 11, Minsk, 220072, Belarus
Email: baranov@im.bas-net.by

Alexander Kleshchev
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403
Email: klesh@math.uoregon.edu

DOI: https://doi.org/10.1090/S0002-9947-99-02003-6
Received by editor(s): November 25, 1996
Additional Notes: Supported by the Fundamental Research Foundation of Belarus and the National Science Foundation
Article copyright: © Copyright 1999 American Mathematical Society