Student seminar

Representation Theory Student Seminar 2021 Semester 1

 

Organisers: Ting Xue, Gufang Zhao

Overview

This is a learning seminar on representation theory, aimed at 3rd year undergraduate students, MSc students, as well as PhD students in mathematics or related fields.

In the learning seminar, participants are expected to learn a topic based on the references given, and present their work to the other participants. The audience are expected to ask questions and make comments during the presentation. Discussions tangential to the topics are welcome. The idea is for all participants to actively learn and discuss together.

This semester, the topic is representation theory of symmetric groups. We assume basic knowledge of group theory and linear algebra. The seminar will begin with basic notions of representations of symmetric groups and symmetric functions, and statements of theorems on characters. Then, we will discuss two different approaches to establish these theorems, Zelevinsky’s approach using Hopf algebras and the Vershik-Okounkov approach.

Time: Wednesdays 13:15-14:15pm

Location: Zoom (Please contact the organisers for the link)

Schedule

3 Mar 13:15-14:15 Zhongtian Chen Universal positive self-adjoint Hopf algebra: uniqueness, continued.

24 Feb Zhongtian Chen Universal positive self-adjoint Hopf algebra: uniqueness, continued. Notes

17 Feb Zhongtian Chen Universal positive self-adjoint Hopf algebra: uniqueness, continued. Notes

10 Feb Zhongtian Chen Universal positive self-adjoint Hopf algebra: uniqueness. Notes

 

 

Representation Theory Student Seminar 2020 Semester 2

 

16 Dec Simon Thomas Hopf algebras, irreducible, and primitive elements, II Notes

9 Dec Simon Thomas Hopf algebras, irreducible, and primitive elements, I Notes

2 Dec Davood Nejaty Frobenius’s formula and applications to topology Notes

28 Oct Yifan Guo Representations of finite groups, characters, symmetric groups, III Notes

21 Oct Yifan Guo Representations of finite groups, characters, symmetric groups, II Notes

14 Oct Yifan Guo Representations of finite groups, characters, symmetric groups, I Notes

30 Sep Organisational meeting

 

Date Topics References Speaker
Oct 14/21/28  Representations of finite groups, characters, symmetric groups [Za] A.1, A.1.2 Yifan Guo
Dec 2  Frobenius’s formula and applications [Za] A.1.3 Davood Nejaty
Dec 9/16 Hopf algebras, irreducible,  and primitive elements [Ze] 1.3- 2. (Definition 1.4, Theorem 2.2 and briefly its proof) Simon Thomas
Feb 10/17/24, 2021 Universal positive self-adjoint Hopf algebra: uniqueness. [Ze] 3 Zhongtian Chen
 Universal positive self-adjoint Hopf algebra: special elements. [Ze] 4 Ennes Mehmedbasic
Symmetric functions,   induction and restriction. [Ze] 5.3, 6 Kshitija Vaidya
Gelfand-Zeitlin algebra and Young-Jucys-Murphy elements. [OV] 1, 2. Weiying Guo
Degenerate affine Hecke algebra and representations. [OV] 3, 4. Adam Monteleone
Branching theorem. [OV] 5, 6. Davood Nejaty
Branching rule and  Murnaghan–Nakayama rule. [OV] 7, 8 tba
References [VO] A. M. Vershik and A. Yu. Okounkov, A New Approach to the Representation Theory of the Symmetric Groups. II, arXiv:math.RT/0503040.

[Za] D. Zagier, Applications of the representation theory of finite groups, appendix to Graphs on Surfaces and Their Applications, (2004).

[Ze] A. Zelevinsky, Representations of Finite Classical Groups: A Hopf Algebra Approach, (1981).