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A note on the Grothendieck ring of the symmetric group. (English) Zbl 1102.20009

Summary: Let \(p\) be a prime number and let \(n\) be a non-zero natural number. We compute the descending Loewy series of the algebra \({\mathcal R}_n/p{\mathcal R}_n\), where \({\mathcal R}_n\) denotes the ring of virtual ordinary characters of the symmetric group \(\mathcal S_n\).

MSC:

20C30 Representations of finite symmetric groups
19A22 Frobenius induction, Burnside and representation rings
05E10 Combinatorial aspects of representation theory
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References:

[1] Curtis, C. W.; Reiner, I., Methods of Representation Theory, vol. I, With Applications to Finite Groups and Orders, Wiley Classics Library (1990), A Wiley-Interscience Publication, John Wiley & Sons, Inc.: A Wiley-Interscience Publication, John Wiley & Sons, Inc. New York, (Reprint of the 1981 original)
[2] Geck, M.; Pfeiffer, G., Characters of Finite Coxeter Groups and Iwahori-Hecke Algebras, London Math. Soc. Monogr. (N.S.), vol. 21 (2000), The Clarendon Press, Oxford University Press: The Clarendon Press, Oxford University Press New York · Zbl 0996.20004
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