Group representations 1, LS 2022/2023
Where and when
Lectures: Mondays 9:00 - 10:30, K9
Problem sessions: Mondays 10:40 - 12:10, K9
Exam and homeworks
The final exam consists of 2 theoretical questions and one question requiring either to compute example
or to find an easy proof.
During the course 4 - 5 sets of homeworks will be posed. You have to solve 3 of them. The deadline for homeworks is August 31.
Recordings
Audio presentations from 2019/2020 should be available
here .
Homeworks
#1 Representations of dihedral groups over C.
#2 Alternative approach to the decomposition of a regular representation.
#3 Complex and real representations.
#4 Normal basis in characteristic 0.
#5 Representations of symmetric groups.
Contents of the lectures
February 13: Basic notions - linear representations, matrix representations, equivalent representations.
Example representations of 2-element group over a field of characteristic different from 2. Presentation (LS 2020 / 2021)
Problem sessions (solutions)
February 20: Group algebras, linear representations as modules over group algebras. Example - representations of
a 3-element group over rational numbers. Presentation (LS 2020 / 2021)
Problem sessions (solutions)
February 27: Example - representations of Z_2 over field of characteristic two. Invariant subspaces, representations restircted to
an invariant subspace and the representation on the factor space. Irreducible representations. Maschke's theorem.
Presentation (LS 2020 / 2021).
Problem sessions (solutions)
March 6: Wedderburn-Artin theorem on structure of semisimple artinian rings. Example which should clarify how to find all irreducible representations
of a finite group over a field of complex numbers one we know the isomorphism between the group algebra and the corresponding product of matrix rings.
Example: Representations of Z_n over complex numbers found using the Chinese reminder theorem. Partially covered by presentations from LS 2020/2021
here and here .
Problem sessions (solutions)
March 13: Representations given by actions of a group on a set, regular representation. Krull-Schmidt theorem for direct sum of simple modules
and the multiplicity. Decompositions of a regular representation of a finite group over an algebraically closed fields of nice characteristics -
number of irreducible representations, multiplicity = dimension, sum of squares of degrees of irreducible representations = the order of the group.
Example - complex represenations of S_3. Handwritten notes (probably LS 2019 / 2020).
Problem sessions (solutions)
March 20: Representations of finite abelian groups over algebraically closed fields of suitable characteristics.
Schur's lemma for modules, some details in the proof of the Wedderburn-Artin theorem. Related notes from LS 2019/2020
are here and here.
Problem sessions (solutions)
March 27: Character theory - definititions and basic properties. Summary of various versions of Schur's lemma
Presentation (LS 2020 / 2021), handwritten notes (LS 2019/2020) are available
here .
Problem sessions (solutions)
April 3: Orthogonality of characters, aplications how to recognize irreducibility and equivalence for representations
of a finite group of characteristic zero.
Presentation, part I (LS 2020 / 2021), part II.
Handwriten notes (LS 2019/2020) are here and here .
Problem sessions (solutions) - character table of A5
April 10: No lecture.
April 17: Character table of a finite group.
Presentation (LS 2020 / 2021). Handwritten notes (including also part of the
next lecture) are here .
The prepared set of problems did not match the pace of the lecture.
April 24: Structural constants of Z(CG) and the complex character table of G. Lecture: How to compute structural constants from the character table.
Problem session: How to compute character table from structural constants. The degree theorem - proof based on basic commutative algebra (lecture) and
a central idempotents in CG (problem sessions). Presentation (LS 2020/2021),
handwritten notes .
Problem sessions (solutions) , other approach to the degree theorem is covered in
handwritten notes from LS 2019/2020
May 1: No lecture
May 8: No lecture
May 15: Burnside's p^aq^b theorem. Presentation (LS 2020/2021) and handwritten notes .
Problem session: Some other results from group theory seen as a consequences of results on group representations
(presentation).
Some remarks on extension of scalars.
May 22: Complex representations of the Symmetric group (2 lectures), part 1 and part 2 .
Handwritten notes are available part 1, part 2
Info: The page will be completed by May 28. If you want to take an exam, please send me an email. You do not need to submit a homework before taking the exam.