2021 Advanced topics in Algebra A1

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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.A405
Credits
1
Academic year
2021
Offered quarter
1Q
Syllabus updated
2021/4/12
Lecture notes updated
-
Language used
English
Access Index

Course description and aims

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.

Student learning outcomes

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 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 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.

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.

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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