Synopsis for Representation Theory of the Symmetric Groups

Recommended Prerequisites

A thorough knowledge of linear algebra and the second year algebra courses; in particular familiarity with the symmetric groups, (symmetric) group actions, quotient vector spaces, isomorphism theorems and inner product spaces will be assumed. Some familiarity with basic representation theory from B2 (group algebras, simple modules, reducibility, Maschke's theorem, Wedderburn's theorem, characters) will be an advantage.

Overview

The representation theory of symmetric groups is a special case of the representation theory of finite groups. Whilst the theory over characteristic zero is well understood, this is not so over fields of prime characteristic. The course will be algebraic and combinatorial in flavour, and it will follow the approach taken by G. James. One main aim is to construct and parametrise the simple modules of the symmetric groups over an arbitrary field. Combinatorial highlights include combinatorial algorithms such as the Robinson–Schensted–Knuth correspondence. The final part of the course will discuss some finite-dimensional representations of the general linear group , and connections with representations of symmetric groups. In particular we introduce tensor products, and symmetric and exterior powers.

Synopsis

Counting standard tableaux of fixed shape: Young diagrams and tableaux, standard-tableaux, Young–Frobenius formula, hook formula. Robinson–Schensted-Knuth algorithm and correspondence.
Construction of fundamental modules for symmetric groups: Action of symmetric groups on tableaux, tabloids and polytabloids; permutation modules on cosets of Young subgroups. Specht modules, and their standard bases. Examples and applications.
Simplicity of Specht modules in characteristic zero and classification of simple -module over characteristic zero. Characters of symmetric groups, Murnaghan–Nakayama rule.
Submodule Theorem, construction of simple -modules over a field of prime characteristic. Decomposition matrices. Examples and applications.
Some finite-dimensional -modules, in particular the natural module, its tensor powers, and its symmetric and exterior powers. Connections with representations of over .

1. W. Fulton, Young Tableaux, London Mathematical Society Student Texts 35 (Cambridge University Press, 1997). From Part I and II.
2. D. Knuth, The Art of Computer Programming, Volume 3 (Addison–Wesley, 1998). From Chapter 5.
3. B. E. Sagan, The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, Graduate Texts in Mathematics 203 (Springer–Verlag, 2000). Chapters 1 – 2.

Additional Reading

1. W. Fulton, J. Harris, Representation Theory: A first course, Graduate Texts in Mathematics, Readings in Mathematics 129 (Springer–Verlag, 1991). From Part I.
2. G. James, The Representation Theory of the Symmetric Groups, Lecture Notes in Mathematics 682 (Springer–Verlag, 1978).
3. G. James, A. Kerber, The Representation Theory of the Symmetric Groups, Encyclopaedia of Mathematics and its Applications 16, (Addison–Wesley, 1981). From Chapter 7.
4. R. Stanley, Enumerative Combinatorics. Volume 2, Cambridge Studies in Advanced Mathematics 62 (Cambridge University Press, 1999).