Syllabus
1. Summary of some results from algebraic number theory and commutative algebra. Tensor product of algebras.
2. Further results on induced representations. Theorem of Artin and Brauer giving expressions of a character as a combination of induced characters. Any complex representation of a finite group G is defined over the exp(G)-th cyclotomic field.
3. Very basic methods from modular representation theory. Composition series, Jacobson radical, finite representation type. Brauer characters and their relation to composition series. Blocks.
4. Integral representations of finite groups. Lattices, the notion of finite representation type for integral representation. Representation type of cyclic groups. The relation between K_0(Z[C_n]) and the ideal class group of cyclotomic integers. Integral representations from the point of view of the representation theory of artin algebras, Klinger-Levy programme.
5. Local-global methods in integral representation theory. Jordan-Zassenhaus theorem, genus. Projective modules over Z[G], a theorem of Swan. Induced indecomposable representations, Green correspondence.
6. (only informatively) Some aspects of representation theory of compact groups or the theory of maximal orders.
Annotation
The course gives a brief overview of some classical results on modular and integral representations of finite groups.