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Unequal Characteristic: Simple Modules, Decomposition Matrices

  • Cédric BonnaféEmail author
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Part of the Algebra and Applications book series (AA, volume 13)

Abstract

In this chapter we will determine the simple kG-modules and the decomposition matrix \(\mathrm{Dec}(\mathcal{O}G)\), as a function of the prime number . This study will be carried out block by block, or more precisely by type of block (nilpotent, quasi-isolated, principal). When the defect group is cyclic we will also give the Brauer tree. We refer the reader to Appendix B for the definitions of these concepts.

To carry out this study, we will use the equivalences of categories constructed in the previous chapter. When these equivalences are Morita equivalences the simple modules correspond to one another and the decomposition matrices are preserved (as is the Brauer tree). When the equivalences in question are genuine Rickard equivalences (which only occurs for the principal block when is odd and divides q+1) the only property that is conserved is the number of simple modules.

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Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.Institut de Mathématiques et de Modélisation de MontpellierCNRS (UMR 5149), Université Montpellier 2Montpellier CedexFrance

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