Syllabus

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