chapter  5
Representation Theory
Pages 14

Our intention in this chapter is to briefly recall the essential notions and results of representation theory of finite groups, especially in the case of the symmetric group. Following Frobenius, we bijectively associate a symmetric function to characters of representations of Sn. It turns out that, under this natural passage to symmetric functions, Schur functions correspond to irreducible representations. It follows that symmetric functions associated with representations are Schur-positive, with coefficients corresponding to multiplicities of irreducible representations. This will be a recurring theme throughout the rest of this book. For proofs of the results outlined here, we refer to [Sagan 91], or the now classic [Fulton and Harris 91]. Many other notions not presented here are found in [Goodman and Wallach 98].