Representation Theory of Finite Groups

Week 1: Review of Linear algebra: Inner product spaces, diagonalizations.
Week 2:Representation theory: an introduction through basic linear algebra. Representations of finite cyclic groups.
Week 3:Representations of algebras: definitions, examples, indecomposable/decomposable representations, reducible/irreducible representations.
Week 4:Representation of finite groups: Maschke’s Theorem on complete reducibility, Schur’s lemma.
Week 5:Character theory: orthogonality of irreducible characters, class functions
Week 6:Induced characters and Frobenius reciprocity
Week 7:The regular representation. Decomposition of regular representation. Number of irreducible representations.
Week 8:The complex function space on G, orthogonality of co-efficient functions.
Week 9:Tensor product of vector spaces, tensor product of representations, sym and alt constructions
Week 10:Dimension theorem, Burnside’s theorem. Induced Representations and Frobenius reciprocity
Week 11:Mackey’s Irreducibility Criterion
Week 12:Representation theory of the symmetric group: the Specht modules of partitions of n