Vol. 3, No. 7, 2009

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A 2-block splitting in alternating groups

Christine Bessenrodt

Vol. 3 (2009), No. 7, 835–846
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. In a previous paper, the first explicit block splitting of regular classes for a family of groups was given for the 2-regular classes of the symmetric groups. Based on this work, the corresponding splitting problem is investigated here for the 2-regular classes of the alternating groups. As an application, an easy combinatorial formula for the elementary divisors of the Cartan matrix of the alternating groups at p = 2 is deduced.

Keywords
alternating groups, $p$-regular conjugacy classes, irreducible characters, Brauer characters, $p$-blocks, Cartan matrix
Mathematical Subject Classification 2000
Primary: 20C15
Secondary: 20C20, 20C30
Milestones
Received: 9 December 2008
Revised: 4 August 2009
Accepted: 5 August 2009
Published: 29 November 2009
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/