Representations of Symmetric Groups

Abstract

This chapter gives an overview of the representation theory of symmetric groups. We start with the characteristic 0 theory. The hook length formula gives the irreducible character degrees for symmetric groups. By contrast, the irreducible Brauer character degrees are not known. The branching rule describes the induction of ordinary characters of Sn−1 to Sn, and again in characteristic p things are much more complicated. We then switch to characteristic p, where we give the construction of Specht modules over \(\mathbb {Z}\), and then look at the block structure and what is known about decomposition numbers. Finally we summarize what is known about the double covers of the symmetric groups.