Representations of groups 1 2023/24

David Stanovský //

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)

Auxiliary materials:

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):

Q5

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
30.4. DS
ZP Application: Degree theorem, Burnside's pq theorem
Example: Representations of Sn St. 6
St. 7, Sen. 6
7.5. ZP Example: Representations of Sn St. 7, Sen. 6
14.5. -- --- volno ---
21.5. ZP Fourier transform on groups ??? (St. 5 is perhaps not the best source)

Zápočet: At least 60% points from quizzes, posted once in two weeks.
Table with results.

Exam: Combination of written (computational problems) and oral (theoretical questions) exam. Details later.