Math 739: Topic Class in Representation Theory

*Lecture notes*

- Preliminary definitions: representations, equivalence, construction of new representations (notes)
- Invariant sub-spaces, irreducible representations, Shur's lemma (notes)
- Indecomposable representations, Mascke's theorem (notes)
- Complete reducibility of representations, decomposition of regular representation, representations of the symmetric group
*S_3*(notes) - Characters (definition, properties, examples); conjugacy classes (dihedral group, symmetric group, group of even permutations) (notes: part 1 and part 2)
- Orthogonality of characters (notes)
- The number of irreducible characters (notes)
- The number of linear characters; the character table of the symmetric group
*S_4*(notes) - Various problems on character tables (notes: part 1 and part 2)
- Restriction of characters: theory and examples (notes)
- Induced representations (notes)
- Frobenius character formula (notes)
- Additional results on restricted and induced representations; Mackey's irreduciblity criterion; semi-direct product by an abelian normal subgroup (notes: part 1 and part 2)
- Semi-direct product by an abelian normal subgroup (continued) (notes)
- Representations of the symmetric group: an introduction (notes)
- Specht modules (notes)
- Ordering of partitions; classification of irreducible representations of
*S_n*(notes) - Bases of Specht modules (notes)
- Branching rules; ring of symmetric functions, monomial and elementary symmetric functions (notes)
- Complete symmetric functions and Shur's functions (notes: part 1 and part 2)
- Alternative definitions of Shur's functions; Cauchy's formula (notes: part 1, part 2 and part 3)
- Power sums; the connection between symmetric functions and irreducible representations of the symmetric group; Frobenius character formula (notes: part 1, part 2 and part 3)
- The ring of representation of
*S_n*and the ring of symmetric functions (notes)