Title On the Representation of the Symmetric Group

Olivier Mathieu

The classical approach of the representation theory of the symmetric group S_{n} is as follows. Each Young diagram Y provides a Shur module ∇_{Y}. Then the simple representations of the symmetric groups are obtained by using Gram-Schmidt orthogonalisation process to the family of modules ∇_{Y}. Each step is very elementary, but the combinatorics is difficult to follow (for example, the set of Young diagram is only a poset).

By contrast, representation theory of compact Lie groups is more advanced, but the theory, including Weyl character formula, is much more easy to understand. In the course, we will use the Schur Weyl duality. With this approach, representation theory for the symmetric group is as a natural consequence of the theory for the groups SU(n).

If time allows, we will also discuss modular representations of S_{n}.