Group Representations

Representation as linear action. Existence of invariant metric. Maschke theorem. Schur's lemma. Multiplicity of irreducible component in regular representation. Structure of group algebra. Characters and presentations of abelian groups, duality. Characters of non-abelian groups. Central idempotent and the first orthogonality relation. Completeness of the set of characters and the second orthogonality relation. Induced representations and characters. Mackey criterion of irreducibility. Representations of symmetric group. Representations of supersolvable groups. Real representations and Schur's index. Other applications of induced representations. Degrees of irreducible representations. Compact groups, Haar measure. Linear representations of compact groups. Unitary trick. Representations of general linear and unitary groups.