2021 Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Algebra A1
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Satoshi Naito
- Class Format
- Lecture
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 5-6 Thu
- Class
- -
- Course Code
- MTH.A405
- Number of credits
- 100
- Course offered
- 2021
- Offered quarter
- 1Q
- Syllabus updated
- Jul 10, 2025
- Language
- English
Syllabus
Course overview and goals
A group representation on a vector space is a group homomorphism from a group to the group of invertible transformations on a vector space.
In this course, we explain the definition and basic properties of group representations, and then explain the definition and basic properties of group characters.
The aim of this course is to explain fundamental facts in the representation theory of finite groups.
Course description and aims
The goal of this course is to be able to write down explicitly character tables of some groups of small order, such as symmetric groups and dihedral groups.
Keywords
finite group, symmetric group, representation, character, character table
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
Standard lecture course.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Definition and examples of groups | Details will be provided during each class session. |
| Class 2 | Definition and examples of group representations | Details will be provided during each class session. |
| Class 3 | Complete reducibility of group representations | Details will be provided during each class session. |
| Class 4 | Schur's lemma on group representations | Details will be provided during each class session. |
| Class 5 | Commutants of group representations | Details will be provided during each class session. |
| Class 6 | Definition and examples of group characters | Details will be provided during each class session. |
| Class 7 | Basic properties of group characters | Details will be provided during each class session. |
| Class 8 | Character relations of group characters | Details will be provided during each class session. |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 30 minutes preparing for class and another 30 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
None.
Reference books, course materials, etc.
Bruce E. Sagan, The Symmetric Group, GTM, No. 203, Springer.
Evaluation methods and criteria
Based on evaluation of assignments. Details will be announced during each class.
Related courses
- MTH.A203 ： Introduction to Algebra III
- MTH.A204 ： Introduction to Algebra IV
- MTH.A301 ： Algebra I
- MTH.A302 ： Algebra II
- MTH.A201 ： Introduction to Algebra I
- MTH.A202 ： Introduction to Algebra II
Prerequisites
None.