A survey of modular representation theory I: Brauer charactersShawn Baland, Aberdeen |
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October 15, 2009 AbstractOne of the most powerful tools used in the representation theory of finite groups is the idea of a group character. When the underlying field k is the complex numbers, we are given an ordinary character table whose rows are indexed by the irreducible characters of G, whose columns are indexed by the conjugacy classes of G, and whose entries are the respective irreducible character values. But when the characteristic of k is a prime dividing |G|, we are forced to take a different approach to character theory in order to obtain results analogous to those in the ordinary case. In this lecture, I will introduce the concepts of representation rings, Grothendieck groups, and Brauer characters, which will allow us to introduce the Brauer character table. Finally, I will use these structures to prove a theorem of Brauer that relates the number of irreducible kG-modules to the number of conjugacy classes in G whose elements have order coprime to the field characteristic. |