### 2021　Advanced courses in Algebra B

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Academic unit or major
Mathematics
Instructor(s)
Naito Satoshi
Course component(s)
Lecture    (ZOOM)
Day/Period(Room No.)
Thr5-6(H137)
Group
-
Course number
ZUA.A332
Credits
1
2021
Offered quarter
2Q
Syllabus updated
2021/3/19
Lecture notes updated
-
Language used
English
Access Index

### Course description and aims

This course is the continuation of "Advanced courses in Algebra A".
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.

### Student learning outcomes

The goal of this course is to understand how the regular representation of a finite group (on its group algebra) decomposes into irreducible representations.

### Keywords

tensor product representation, regular representation, induced representation, Frobenius reciprocity

### Competencies that will be developed

 ✔ Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

### Class flow

Standard lecture course

### Course schedule/Required learning

Course schedule Required learning
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

### Out-of-Class Study Time (Preparation and Review)

To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.

None required

### Reference books, course materials, etc.

Bruce E. Sagan, The Symmetric Group, GTM, No. 203, Springer.

### Assessment criteria and methods

Course scores are evaluated by homework assignments. Details will be announced during the course.

### Related courses

• MTH.A201 ： Introduction to Algebra I
• MTH.A202 ： Introduction to Algebra II
• MTH.A203 ： Introduction to Algebra III
• MTH.A204 ： Introduction to Algebra IV
• MTH.A301 ： Algebra I
• MTH.A302 ： Algebra II

### Prerequisites (i.e., required knowledge, skills, courses, etc.)

linear algebra and basic undergraduate algebra