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 VershikOkounkov approach.
Time: Wednesdays 13:1514:15pm
Location: Zoom (Please contact the organisers for the link)
Schedule
3 Mar 13:1514:15 Zhongtian Chen Universal positive selfadjoint Hopf algebra: uniqueness, continued.
24 Feb Zhongtian Chen Universal positive selfadjoint Hopf algebra: uniqueness, continued. Notes
17 Feb Zhongtian Chen Universal positive selfadjoint Hopf algebra: uniqueness, continued. Notes
10 Feb Zhongtian Chen Universal positive selfadjoint 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 selfadjoint Hopf algebra: uniqueness.  [Ze] 3  Zhongtian Chen 
Universal positive selfadjoint Hopf algebra: special elements.  [Ze] 4  Ennes Mehmedbasic  
Symmetric functions, induction and restriction.  [Ze] 5.3, 6  Kshitija Vaidya  
GelfandZeitlin algebra and YoungJucysMurphy 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).
