Representation Theory of the Symmetric Group

Instructor: Joel Kamnitzer,
Meetings: 1 - 4 pm, BA 4010 (to be confirmed)
Office: 6110 Bahen 416-978-5163
Office Hours: Tuesday 2-3 pm, Wednesday 1-2 pm

All further material will appear on the wiki.

The theme of our course will be the representation theory of the symmetric group. This is a classical subject in algebra which was first developed at the beginning of the 20th century. It has deep connections with algebraic combinatorics, algebraic geometry, and mathematical physics. Though it is an old subject, it continues to be an area of active research. In the fall semester of our course, we will study the foundations of the subject following the book by Sagan. In the winter semester, we will study more recent developments by reading research articles and monographs.

Each week, starting in October, two students will jointly give a two hour presentation on the assigned material. You can split up the time as you see fit, but each student should speak for about an hour. In the fall, the students will have to present the material in their section. In the winter, there will be more flexibility. Giving a math talk is not easy, so please begin preparing your presentation well in advance. Two weeks before your presentation, you should already understand the material well and one week before your presentation, you should have already planned your talk. It is mandatory to come see me during office hours (or by appointment) more than a week before your presentation.

Grading scheme
I anticipate that each student will do four presentations during the year. You will be marked on these presentations according to the following criteria:
Preparedness during initial meeting (25 %)
Knowledge of subject matter during presentation (50 %)
Ability to communicate the material (25%)
Though the presentations are joint, you will be marked individually.

Your final mark will be calculated as follows:
Presentations (60 %)
Assignments (20 %)
Participation (20 %)
The participation is based on attending and and asking questions at the other presentations.

References for the fall semester
Main text: Bruce Sagan, The Symmetric Group: Representations, Combinatorial Algorithms and Symmetric Functions
Other useful references:

William Fulton, Young Tableaux
William Fulton and Joseph Harris, Representation theory: a first course
J. P. Serre, Linear representations of finite groups
Constantin Teleman, Representation theory
Peter Webb, Representation theory for the pure mathematician

Schedule with sources, the main reference is Sagan (S)
September: background lectures
October 1: Groups, representations, and Mashke's Theorem (S 1.1-1.5, FH 1.1-1.2)
October 8: Schur's Lemma, Characters (S 1.6-1.9, FH 1.3, 2.1-2.3)
October 15: Group algebras, induced representations (S 1.10-1.12, FH 3.3-3.4)
October 22: Tableaux, Specht modules (S 2.1-2.3, F 7.1-7.2)
October 29: Irreps, bases, Garnir elements (S 2.4-2.7, F 7.2,7.4)
November 5: Branching, Kostka numbers (S 2.8-2.11, F 7.3)
November 12: Robinson-Schensted, jeu de Taquin (S 3.1-3.7, F 1-4)
November 19: Symmetric functions (S 4.1, 4.3-4.5, F 6)
November 26: Character map, Littlewood Richardson (S 4.7, 4.9, F 7.3)

There will be 2-3 assignments during the fall.

Papers/monographs for the spring semester (incomplete list so far):
A.Vershik, A.Okounkov, A New Approach to the Representation Theory of the Symmetric Groups.
A. Kleshchev, "Branching Rules for Modular Representations of Symmetric Groups I"