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:
Auxiliary materials:
Program: (actual for past lectures, tentative for future lectures):
Q5
date 
lecturer 
topic 
materials 
quizz 
20.2. 
DS 
Introduction: motivation, equivalence, regular representation. 
Sch. 2.12.7 

27.2. 
ZP 
Maschke's theorem, irreducible representations. 
Sch. 2.82.10 
Q1 till 12.3. 9:00 
5.3. 
DS 
Schur's lemma. Representations of abelian groups and 1dimensional representations. 
Sch. 2.112.13 
12.3. 
ZP 
Representations on Hom(V,W). Uniqueness of irreducible decompositions. Decomposition of regular representation, finding all irreducible representations. 
Sch. 2.142.17 
Q2 till 26.3. 9:00 
19.3. 
DS 
More examples. The number of irreducible representations. 
Sch. 2.172.18 
26.3. 
DS ZP 
Tensor products of representations (quick overview). Characters: definition and basic properties. 
Sch. 2.192.20 Sch. 3.13.3 
Q3 solution 
2.4. 
ZP 
Characters: inner product and dimension of homomorphism space. 
Sch. 3.43.5 
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 
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 
St. 10 
14.5. 
 
 volno  

21.5. 
ZP 
Example: Representations of Sn. 
St. 10 
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
All topics covered by the lecture / exercises can appear at the exam, with two exceptions: tensor products (the theory of sections 2.1920), and the moduletheoretic approach. However, if you want to continue in any of the followup courses (Representations 2, or any course in abstract representation theory), or if you choose representations as your topic for the state exam, you are expected to understand the correspondence.
