The 2-block splitting in symmetric groups

Bessenrodt, Christine

doi:10.2140/ant.2007.1.223

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The 2-block splitting in symmetric groups

Christine Bessenrodt

Vol. 1 (2007), No. 2, 223–238

DOI: 10.2140/ant.2007.1.223

Abstract

In 1956, Brauer showed that there is a partitioning of the $p$ -regular conjugacy classes of a group according to the $p$ -blocks of its irreducible characters with close connections to the block theoretical invariants. But an explicit block splitting of regular classes has not been given so far for any family of finite groups. Here, this is now done for the 2-regular classes of the symmetric groups. To prove the result, a detour along the double covers of the symmetric groups is taken, and results on their 2-blocks and the 2-powers in the spin character values are exploited. Surprisingly, it also turns out that for the symmetric groups the 2-block splitting is unique.

Keywords

symmetric groups, $p$-regular conjugacy classes, Cartan matrix, irreducible characters, Brauer characters, $p$-blocks, spin characters

Mathematical Subject Classification 2000

Primary: 20C30

Secondary: 20C15, 20C20

Milestones

Received: 3 May 2007

Revised: 2 July 2007

Accepted: 8 August 2007

Published: 1 May 2007

Authors

Christine Bessenrodt
Institut für Algebra, Zahlentheorie und Diskrete Mathematik Fakultät für Mathematik und Physik Leibniz Universität Hannover Welfengarten 1 D-30167 Hannover Germany
http://www-ifm.math.uni-hannover.de/~bessen/

Vol. 1, No. 2, 2007