Vol. 1, No. 2, 2007

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The 2-block splitting in symmetric groups

Christine Bessenrodt

Vol. 1 (2007), No. 2, 223–238

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.

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
Received: 3 May 2007
Revised: 2 July 2007
Accepted: 8 August 2007
Published: 1 May 2007
Christine Bessenrodt
Institut für Algebra, Zahlentheorie und Diskrete Mathematik
Fakultät für Mathematik und Physik
Leibniz Universität Hannover
Welfengarten 1
D-30167 Hannover