2025 (Current Year) 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 (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 5-6 Thu
- Class
- -
- Course Code
- MTH.A406
- Number of credits
- 100
- Course offered
- 2025
- Offered quarter
- 2Q
- Syllabus updated
- Mar 19, 2025
- Language
- English
Syllabus
Course overview and goals
The representation theory of groups investigates the different ways in which a given group acts on vector spaces.
In this lecture, we first explain the classification theory of the (finite-dimensional) irreducible representations of symmetric groups over the complex numbers, and then give an explicit realization of irreducible representations.
Also, we explain how to compute explicitly the characters of irreducible representations.
For this purpose, rather than following the classical route by Frobenius, Schur, and Young, we take an elegant, novel approach devised by Okounkov-Vershik.
This course is based on "Advanced topics in Algebra A1" in the first quarter.
Course description and aims
Students are expected to:
- understand the classification theory of irreducible representations of symmetric groups,
- understand how to realize explicitly irreducible representations of symmetric groups,
- understand how to compute explicitly irreducible characters of symmetric groups.
Keywords
symmetric groups, irreducible representations, irreducible characters, Murnaghan-Nakayama rule
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 | Hook length formula and its proof | Details will be provided during each class session. |
| Class 2 | Graph isomorphism theorem: part 1 | Details will be provided during each class session. |
| Class 3 | Graph isomorphism theorem: part 2 | Details will be provided during each class session. |
| Class 4 | Explicit realization of irreducible representations | Details will be provided during each class session. |
| Class 5 | Gelfand-Zetlin basis of an irreducible representation | Details will be provided during each class session. |
| Class 6 | Irreducible characters for symmetric groups | Details will be provided during each class session. |
| Class 7 | Schur's double centralizer theorem and Schur functors | Details will be provided during each class session. |
| Class 8 | Schur-Weyl duality | 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 required.
Reference books, course materials, etc.
T. Ceccherini-Silberstein, F. Scarabotti, F. Tolli, Representation Theory of the Symmetric Groups, Cambridge University Press, 2010.
M. Lorenz, A Tour of Representation Theory, American Mathematical Society, 2018.
Evaluation methods and criteria
Assignments (100%).
Related courses
- MTH.A201 ： Introduction to Algebra I
- MTH.A202 ： Introduction to Algebra II
- MTH.A301 ： Algebra I
- MTH.A302 ： Algebra II
- MTH.A211 ： Advanced Linear Algebra I
- MTH.A212 ： Advanced Linear Algebra II
- MTH.A405 ： Advanced topics in Algebra A1
Prerequisites
Basic knowledge of advanced linear algebra and some undergraduate algebra, including rings and modules, is preferable.
Also, it is recommended to take "MTH.A405" before taking the current course.