Time and Place:
- Introduction: Actions of groups on sets. Induced linear actions. Multilinear algebra.
- Representations of groups, direct sum. Irreducible representations, semi-simple representations. Schur’s lemma. Irreducible representations of finite abelian groups. Complete reducibility, Machke’s theorem.
- Equivalent representations. Morphisms between representations. The category of representations of a finite group. A description using the group ring. Multilinear algebra of representations: dual representation, tensor product (inner and outer).
- Decomposition of the regular representation into irreducible representations. The number of irreducibles is equal to the number of conjugacy classes. Matrix coefficients, characters, orthogonality.
- Harmonic analysis: Fourier transform on finite groups and the non-commutative Fourier transform.
- Frobenius divisibility and Burnside $p^aq^b$ theorem.
- Constructions of representations: induced representations. Frobenius reciprocity. The character of induced representation. Mackey’s formula. Mackey’s method for representations of semi-direct products.
- Induction functor: as adjoint to restrictions, relation to tensor product. Restriction problems, multiplicity problems, Gelfand pairs and relative representation theory.
- Examples of representations of specific groups: $SL_2$ over finite fields, Icosahedron group, Symmetric groups.
- Artin and Brauer Theorems on monomial representations