2021 Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Algebra B1
- 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.A406
- Number of credits
- 100
- Course offered
- 2021
- Offered quarter
- 2Q
- Syllabus updated
- Jul 10, 2025
- Language
- English
Syllabus
Course overview and goals
This course is the continuation of "Advanced topics in Algebra A1".
A group representation on a vector space is a group homomorphism from a group to the group of invertible linear transformations on a vector space.
The aim of this course is to explain fundamental facts in the representation theory of finite groups;
in particular, we explain tensor product representations, induced representations, and the relationship between restriction and induction of group representations.
Course description and aims
The goal of this course is to understand how the regular representation of a finite group (on its group algebra) decomposes into irreducibles ones.
Keywords
tensor product representation, regular representation, induced representation, Frobenius reciprocity
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 | Regular representation |
Details will be provided during each class session. |
| Class 2 | Irreducible decomposition of the regular representation |
Details will be provided during each class session. |
| Class 3 | Tensor product representations |
Details will be provided during each class session. |
| Class 4 | Representation matrices of tensor product representations |
Details will be provided during each class session. |
| Class 5 | Induced representations |
Details will be provided during each class session. |
| Class 6 | Representation matrices of induced representations |
Details will be provided during each class session. |
| Class 7 | Relationship between restriction and induction of representations |
Details will be provided during each class session. |
| Class 8 | Frobenius Reciprocity |
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 session.
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