The syllabus will updated at the course progresses. An overview of the topics to be covered in the course can be found here.
| Date | Notes | Topics |
|---|---|---|
| Sep 6 | 1.1.1, 1.1.2 | Representations of finite groups |
| Sep 11 | 1.1.3–1.1.5 | Intertwining operators, direct sums, Maschke's Theorem, the adjoint representation |
| Sep 13 | 1.1.6–1.1.8 | Matrix coefficients, tensor products, cyclic and invariant vectors |
| Sep 18 | 1.2.1–1.2.3 | Schur's lemma, isotypic components, finite-dimensional algebras |
| Sep 20 | 1.2.4, 1.2.5 | The commutant, interwiners as invariant elements |
| Sep 25 | 1.3.1–1.3.3 | The trace, central functions, characters, central projection formulas |
| Sep 27 | 1.3.3, 1.4.1 | Central projection formulas, Wielandt's lemma |
| Oct 2 | 1.4.2, 1.4.3 | Symmetric actions, Gelfand's lemma, Frobenius reciprocity for permutation representations |
| Oct 4 | 1.4.4, 1.5.1 | The commutant of a permutation representation, the group algebra |
| Oct 11 | 1.5.1, 1.5.2 | The group algebra, the Fourier transform |
| Oct 16 | 1.5.3, 1.6.1, 1.6.2 | Algebras of bi-K-invariant functions, induced representations |
| Oct 18 | 1.6.3, 1.6.4 | Frobenius reciprocity, Mackey's Lemma, the intertwining number theorem |
| Oct 23–27 | Reading week | |
| Oct 30 | 2.1.1, 2.1.2 | Conjugacy invariant functions, multiplicity-free subgroups |
| Nov 1 | 2.1.2, 2.2.1, 2.2.2 | Branching graphs, Gelfand-Tsetlin bases, Gelfand-Tsetlin algebras |
| Nov 6 | 2.2.2, 3.1.1–3.1.3 | Partitions and conjugacy classes of symmetric groups, Young diagrams, Young tableaux |
| Nov 8 | 1.1.1–2.2.1 | Midterm test |
| Nov 13 | 3.1.4–3.1.6 | Coxeter generators, content of a tableau, the Young poset |
| Nov 15 | 3.2.1–3.2.3 | Young-Jucys-Murphy elements, marked permutations |
| Nov 20 | 3.2.2–3.3.2 | Olshanskii's Theorem, weights, the spectrum of the YJM elements |
| Nov 22 | 3.3.3 | Spectrum versus content |
| Nov 27 | 3.4.1, 3.4.2 | Young's seminormal form, Young's orthogonal form |
| Nov 29 | 3.4.3, 3.4.4, 4.1 | The Young seminormal units, the Theorem of Jucys and Murphy, Schur-Weyl duality |
| Dec 4 | 4.2.1–4.2.3 | Categorification of symmetric functions |
| Dec 6 | 4.2.4–4.2.7 | Categorification of bosonic Fock space and the basic representation |