David Stanovský
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| REPRESENTATIONS OF GROUPS 1 (2023/24) |
Lecturers: Zuzana Patáková, David Stanovský
Syllabus:
- Fundamentals of representation theory: Maschke, Schur, counting representations
- Characters
- Representations of Sn, hook length formula
- Burnside's pq theorem
- Fourier analysis on finite groups
Literature: We will use the following two lecture notes:
- by Travis Schedler (Imperial College) (chapters 2 and 3, the first eight lectures)
- by Benjamin Steinberg (Carleton Uni.) (chapters 5, 6, 7 in the second half of the course)
- materials by Pavel Příhoda from the previous years
- notes by Ambar Sengupta (Louisiana State) (an alternative for the topics in the second half of the course)
- proof of the hook length formula
Program: (actual for past lectures, tentative for future lectures):
| date | lecturer | topic | materials | quizz |
| 20.2. | DS | Introduction: motivation, equivalence, regular representation. | Sch. 2.1-2.7 | |
| 27.2. | ZP | Maschke's theorem, irreducible representations. | Sch. 2.8-2.10 | Q1 till 12.3. 9:00 |
| 5.3. | DS | Schur's lemma. Representations of abelian groups and 1-dimensional representations. | Sch. 2.11-2.13 | |
| 12.3. | ZP | Representations on Hom(V,W). Uniqueness of irreducible decompositions. Decomposition of regular representation, finding all irreducible representations. |
Sch. 2.14-2.17 | Q2 till 26.3. 9:00 |
| 19.3. | DS | More examples. The number of irreducible representations. | Sch. 2.17-2.18 | |
| 26.3. | DS ZP |
Tensor products of representations (quick overview). Characters: definition and basic properties. |
Sch. 2.19-2.20 Sch. 3.1-3.3 |
Q3 solution |
| 2.4. | ZP | Characters: inner product and dimension of homomorphism space. | Sch. 3.4-3.5 | |
| 9.4. | ZP, DS | Character tables and orthogonality, examples. | Sch. 3.6 | Q4 till 23.4.9:00 |
| 16.4. | DS | Character tables: kernel, normal subgroups, automorphisms | Sch. 6.7, 6.8 | |
| 23.4. | DS | Abstraction: modules over group algebras Application: Degree theorem, Burnside's pq theorem |
Sch. 4, Pří. St. 6 |
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| 30.4. | DS | Abstraction: modules over group algebras Application: Degree theorem, Burnside's pq theorem |
Sch. 4, Pří. St. 6 |
Q5 till 21.5. 9:00 |
| 7.5. | ZP | Example: Representations of Sn | --- | |
| 14.5. | -- | --- volno --- | ||
| 21.5. | ZP | Example: Representations of Sn. | --- | Q6 till 4.6. 9:00 |
Zápočet: At least 60% points (=27) from quizzes, posted once in two weeks.
Table with results.
Exam: Combination of written (computational problems) and oral (theoretical questions) exam.
- written test (120 min): computational exercises, theorem statements, short proofs (from the lecture, or similar to those in the lecture)
- oral exam: a discussion of a particular topic, including all proofs