# Alessandra Pantano

Math 739: Topic Class in Representation Theory

Lecture notes

1. Preliminary definitions: representations, equivalence, construction of new representations (notes)
2. Invariant sub-spaces, irreducible representations, Shur's lemma (notes)
3. Indecomposable representations, Mascke's theorem (notes)
4. Complete reducibility of representations, decomposition of regular representation, representations of the symmetric group S_3 (notes)
5. Characters (definition, properties, examples); conjugacy classes (dihedral group, symmetric group, group of even permutations) (notes: part 1 and part 2)
6. Orthogonality of characters (notes)
7. The number of irreducible characters (notes)
8. The number of linear characters; the character table of the symmetric group S_4 (notes)
9. Various problems on character tables (notes: part 1 and part 2)
10. Restriction of characters: theory and examples (notes)
11. Induced representations (notes)
12. Frobenius character formula (notes)
13. 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)
14. Semi-direct product by an abelian normal subgroup (continued) (notes)
15. Representations of the symmetric group: an introduction (notes)
16. Specht modules (notes)
17. Ordering of partitions; classification of irreducible representations of S_n (notes)
18. Bases of Specht modules (notes)
19. Branching rules; ring of symmetric functions, monomial and elementary symmetric functions (notes)
20. Complete symmetric functions and Shur's functions (notes: part 1 and part 2)
21. Alternative definitions of Shur's functions; Cauchy's formula (notes: part 1, part 2 and part 3)
22. 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)
23. The ring of representation of S_n and the ring of symmetric functions (notes)