Representation theory of the symmetric group
 January-April 2014
Course description: The course is aimed towards understanding the representation theory of the symmetric group. We will begin with an introduction to the representation theory for finite groups and will cover only the basic results necessary for our purpose. We will lay more emphasis on the combinatorial and computational aspects of the representations under study.
 
Prerequisite: Algebra III. A thorough knowledge of linear algebra and group theory, in particular the symmetric group and group actions will be assumed.

                                                                         
Lectures:  Monday and Wednesday, 9:10 - 10:25 a.m.
Classroom:
  Lecture Hall 4
Instructor:  Kavita Sutar-Deshpande
Contact:  Office: 403
 phone: 962
 email: ksutar AT cmi DOT ac DOT in
Office Hours:
   Mondays and Fridays: 11:00 a.m. - 12:30 p.m.
 (If you cannot make it to my office hours but wish to meet me, please send me an email to set up an appointment at a mutually convenient time)
Reference books:
  • 'The symmetric group: representations, combinatorial algorithms and symmetric functions' - Bruce Sagan
  • 'Representation theory' - Fulton, Harris
  • 'Young tableaux' - Fulton
Teaching Assistant:   Shraddha Srivastava
Grading:
  30% - Final exam
  25% - Homework
  20% - Midterm  
  20% - Project and presentation  
    5% - Instructor's discretion (based on attendance, in-class performance, punctuality etc.)
  All grades will be uploaded on Moodle.

Course components

The course will consist of in-class lectures during all weeks of the semester.
Homework will be assigned every week (barring only exam weeks).
Each student is expected to work on a project throughout the semester and present it in a half-hour talk towards the end of the semester.
There will be a midterm and a  final examination.

Homework

Homework will be posted on this website every Sunday night and will be due in one week (on the following Monday).
You may pick up the graded homework on Tuesdays.
Project

The project consists of writing an article based on your chosen topic and a 20-minute class presentation. The project schedule is as follows:
20 February: first project report
17 March: second project report
April first week: class presentations begin
20 April: Final report/article submission.

All submission must be typeset in LaTeX.
Attendance

Attendance to classes is not required but absences will be noted. Explanation for longer absences (more than one class) is required.

Warning against copying/plagiarism

Please be warned that I will not tolerate any kind of plagiarism in your work. You are free to search for ideas on the internet and in books but the details and work have to be your own.

According to the Merriam-Webster OnLine Dictionary, to plagiarize means:
Synonyms: copying, appropriation, infringement, piracy, counterfeiting, theft, borrowing.    

 Please visit www.plagiarism.org for more details.
Schedule

Week
 Topic
1 (06/01 - 10/01)
 Introduction to group actions and representations;
irreducible representations and complete reducibility.
Homework 1 (due on 13/01)
2 (13/01 - 17/01)

Character theory.
Homework 2 (due on 20/01)
Holiday on 14 January (Pongal).
3 (20/01 - 24/01)

Character theory.
Lectures 5 and 6.
Homework 3 (due on 27/01)
Project proposal (due on 27/01)
4 (27/01 - 31/01)

Induced and restricted representations.
Homework 4 (due on 03/02)
Deadline for course registration - 31 January
5 (03/02 - 07/02)
 Young subgroups, tabloids, ordering on partitions.
Homework 5 (due on 10/02)
6 (10/02 - 14/02)
 Specht modules.
Homework 6 (due on 17/02)
7 (17/02 - 21/02)
Dominance ordering on tabloids.
No class on Wednesday 19/02.
First project report due on 20/02.
22/02 (Saturday)
 Midterm examination (9:30 a.m. to 12:30 p.m.)
8 (24/02 - 28/02)
Mid-semester exams week.
9 (03/03 - 07/03)
 Straightening algorithm, basis of standard tableaux, dimension of Specht modules.
Homework 7 (due on 10/03)
10 (10/03 - 14/03)
 The branching rule, decomposition of \(M^{\mu}\).
Homework 8 (due on 17/03) 
11 (17/03 - 21/03)
 R-S-K correspondence and some applications.
Homework 9 (due on 24/03)
Second project report due on 17/03.
12 (24/03 - 28/03)

 Combinatorial techniques.
Homework 10 (due on 03/04)
13 (31/03 - 04/04)
 Symmetric functions.
14 (07/04 - 11/04)

 Project presentations week (schedule below).
Homework 11 (due on 20/04)
15 (14/04 - 18/04)

 Representation ring and characters.
Holiday on 18 April (Good Friday).
16 (21/04 - 25/04)
 No classes.
Final project submission due on 20/04.
26/04 (Saturday)
 Final examination (9:30 a.m. to 12:30 p.m.)
17 (28/04 - 02/05) Final exams week.

PROJECT REPORTS:
1) Sliding (Jeu de Taquin) - Arpita
2) Viennot's algorithm - Ravitheja
3) R-S-K correspondence - Dinesh Valluri
4) Symmetric functions - Gautam Gopal
5) Hook formula - Navaneeth, JanakiRaman
6) Littlewood-Richardson rule - Aman Barot
7) Murnaghan-Nakayama rule - Chandranandan



Back to my teaching page