**Instructor: ** Joel Kamnitzer, jkamnitz@math.toronto.edu

**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.

** Overview **

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.

** Presentations **

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:

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.

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"