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 |