Since the pioneering works of Frobenius, Schur, and Young more than a hundred years ago, the representation theory of the symmetric group has developed into a huge area of study, with applications to algebra, combinatorics, category theory, and mathematical physics. In this course, we will cover the representation theory of the symmetric group following modern techniques developed by Vershik, Olshankii, and (Fields medalist) Okounkov.
Using techniques from algebra, combinatorics, and category theory, we will cover the following topics.
- Representation theory of finite groups. We will begin the course with an introduction to the representation theory of finite groups. This will include a discussion of irreducible representations, tensor products, Schur's lemma, characters, permutation representations, group algebras, and Frobenius reciprocity.
- The theory of Gelfand-Tsetlin bases. We will discuss branching rules for representations of symmetric groups and see how such branching rules allow one to obtain particularly nice bases for irreducible representations.
- The Okounkov-Vershik approach. We will discuss the combinatorics of Young tableaux, Jucys-Murphy elements, and the Okounkov-Vershik approach to the representation theory of symmetric groups.
MAT 2141 and MAT 2143 and (MAT 3141 or MAT 3143), or equivalent courses given at other universities. (See here for a description of these courses.)
Students are not required to purchase any textbooks for this course. Below are listed some optional books and papers that students may find useful. The main reference for the course will be the lecture notes.
Books and papers
Ceccherini-Silberstein, Scarabotti, Tolli, Representation theory of the symmetric groups, 2010. ISBN: 9780521118170. This book is well suited to the course. For some portions of the course, we will follow it quite closely.
Kleshchev, Linear and projective representations of symmetric groups, 2009. ISBN-13: 9780521104180. This book is a bit more advanced, but will be useful for some portions of the course.
Py, On the representation theory of the symmetric groups, Journal of Mathematical Sciences, Vol. 129, No. 2, 2005.
Click the above header to download lecture notes for the course. They are a single file that will be updated as the course progresses. They should be complemented by your own class notes. I will often say more in class than is in the lecture notes.
There are two main locations for information regarding the course: