Author(s): Steven V. Sam
Edition: version 2020-02-28
Year: 2020
Language: English
Commentary: Downloaded from https://math.ucsd.edu/~ssam/old/20W-202B/notes.pdf
1. Linear representations of finite groups
1.1. Definitions
1.2. Basic operations
1.3. Irreducible representations
1.4. Characters
1.5. Classification of representations
1.6. Examples
1.7. The group algebra
1.8. Restriction and induction
2. Constructing symmetric group representations
2.1. Partitions
2.2. Tabloids
2.3. Specht modules
2.4. Garnir relations and standard tableaux
3. Symmetric functions
3.1. Definitions
3.2. Monomial symmetric functions
3.3. Elementary symmetric functions
3.4. The involution
3.5. Complete homogeneous symmetric functions
3.6. Power sum symmetric functions
3.7. A scalar product
4. Schur functions and the RSK algorithm
4.1. Semistandard Young tableaux
4.2. RSK algorithm
4.3. Dual RSK algorithm
4.4. Determinantal formula
4.5. Multiplying Schur functions, Pieri rule
4.6. Jacobi–Trudi identity
5. Representation theory of the symmetric groups
5.1. Symmetric groups
5.2. The characteristic map
5.3. Murnaghan–Nakayama rule
6. Combinatorial formulas
6.1. Standard Young tableaux
6.2. Semistandard Young tableaux
6.3. Littlewood–Richardson coefficients
7. Polynomial functors
7.1. Basic multilinear algebra
7.2. Schur functors
7.3. Polynomial representations and characters
7.4. Re-interpreting symmetric function identities
References
