Goals of the Course(JPN)
Goals of the Course
|Introduce some basics of the representation theory of finite groups in both ordinary and modular characteristics, including basic understandings of character theoretic, module theoretic, and quiver representation theoretic approaches.
Objectives of the Course(JPN))
Objectives of the Course
|-Understand basic properties of group algebras.
-Be able to translate between representations, characters, and modules.
-Learn to calculate ordinary characters of finite groups of small orders.
-Be able to calculate with quiver-and-relations of certain block algebras, including the Brauer tree algebras, Kronecker algebra and its self-injective analogue.
Course Content / Plan
|Part I: Ordinary characteristic theory:
Representations of groups, Maschke's theorem, tensor and dual, induction and restriction, permutation representations
Basics of ordinary character theory
Part II: Modular characteristic theory:
Artin-Wedderburn theorem, Jordan-Holder theorem, radical/socle series.
Blocks, symmetricity. The group algebras of cyclic groups. Brauer tree algebras. Kronecker algebra and representation-finite groups algebras.
If time allows: Cellular algebra approach for Weyl groups.
|Linear algebra is necessary. Basic knowledge of groups, rings, and modules.
Course Evaluation Method and Criteria
|For Part I:
G. James & M. Liebeck: Representations and characters of groups 2nd ed, Cambridge University Press 2001
For Part II:
D. J. Benson: Representations and Cohomology: Volume 1, Basic Representation Theory of Finite Groups and Associative Algebras, Cambridge Studies in Advanced Mathematics 30, Cambridge University Press 1998
A. Zimmermann: Representation Theory: A Homological Algebra Point of View, Algebra and Applications 19, Springer 2014
Study Load(Self-directed Learning Outside Course Hours)
Notice for Students
Propriety of Other department student's attendance
Conditions of Other department student's attendance
|None, but see prerequisite.
|Reading James and Liebeck will help you understand everything in Part I.
Lecture format, etc.
|Depending on COVID situation, classes may be switchd to online.
Please get in touch as soon as possible if you have trouble attending offline.
Additional measures for remote class (on-demand class)