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Primitive central idempotents of finite group rings of symmetric groups


Author: Harald Meyer
Journal: Math. Comp. 77 (2008), 1801-1821
MSC (2000): Primary 20C05, 20C30, 20C40
DOI: https://doi.org/10.1090/S0025-5718-07-02058-3
Published electronically: December 17, 2007
MathSciNet review: 2398795
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Abstract: Let $ p$ be a prime. We denote by $ S_n$ the symmetric group of degree $ n$, by $ A_n$ the alternating group of degree $ n$ and by $ {\mathbb{F}}_p$ the field with $ p$ elements. An important concept of modular representation theory of a finite group $ G$ is the notion of a block. The blocks are in one-to-one correspondence with block idempotents, which are the primitive central idempotents of the group ring $ {\mathbb{F}}_q G$, where $ q$ is a prime power. Here, we describe a new method to compute the primitive central idempotents of $ {\mathbb{F}}_q G$ for arbitrary prime powers $ q$ and arbitrary finite groups $ G$. For the group rings $ {\mathbb{F}}_p S_n$ of the symmetric group, we show how to derive the primitive central idempotents of $ {\mathbb{F}}_p S_{n-p}$ from the idempotents of $ {\mathbb{F}}_p S_n$. Improving the theorem of Osima for symmetric groups we exhibit a new subalgebra of $ {\mathbb{F}}_p S_n$ which contains the primitive central idempotents. The described results are most efficient for $ p = 2$. In an appendix we display all primitive central idempotents of $ {\mathbb{F}}_2 S_n$ and $ {\mathbb{F}}_4 A_n$ for $ n \le 50$ which we computed by this method.


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Additional Information

Harald Meyer
Affiliation: Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany
Email: harald.meyer@uni-bayreuth.de

DOI: https://doi.org/10.1090/S0025-5718-07-02058-3
Keywords: Group ring, symmetric group, primitive central idempotent
Received by editor(s): December 15, 2006
Received by editor(s) in revised form: March 8, 2007
Published electronically: December 17, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.