2021 Advanced topics in Algebra B1

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Academic unit or major
Graduate major in Mathematics
Instructor(s)
Naito Satoshi 
Course component(s)
Lecture    (ZOOM)
Day/Period(Room No.)
Thr5-6()  
Group
-
Course number
MTH.A406
Credits
1
Academic year
2021
Offered quarter
2Q
Syllabus updated
2021/4/12
Lecture notes updated
-
Language used
English
Access Index

Course description and aims

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.

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 irreducibles ones.

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.
Class 8 Frobenius Reciprocity 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 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.

Assessment criteria and methods

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

None

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