Unequal Characteristic: Simple Modules, Decomposition Matrices

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 ℓ. 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 ℓ is odd and divides q+1) the only property that is conserved is the number of simple modules.