Chapter 2 - Ordinary Irreducible Representations and Characters of Symmetric and Alternating Groups

Summary

The present chapter contains a characterization of the ordinary irreducible representations of symmetric groups as common constituents of monomial representations induced from Young subgroups. Besides this, the ordinary irreducible characters are given as ℤ-linear combinations of permutation characters induced from Young subgroups, and an important recursion formula is derived. As an application, the ordinary irreducible characters and representations of A_n are given. It is shown furthermore, that S_n is characterized by its character table. An examination of the values of the irreducible characters leads to the notions of cores and quotients of partitions which later on are of great importance for modular purposes. Finally, the Littlewood-Richardson rule, which is one of the most useful results of the whole theory, is derived.

The Ordinary Irreducible Representations of S_n

By an ordinary representation of a group G we mean a finite-dimensional representation of G over the field ℂ of complex numbers. It will turn out that each ordinary representation of S_n can be realized over the rational field, so that ℚ (and thus every field) is a splitting field for S_n. Hence it does not make much difference whether we assume that the groundfield F is of a characteristic which does not divide n! or we just consider the ordinary representations of S_n.

We know from general representation theory of finite groups that S_n has the same number of ordinary irreducible representations as conjugacy classes.