7 - Representations of the symmetric group

Summary

The first section describes the action of the symmetric group S_n on the numberings of a Young diagram with the integers 1, 2, …, n with no repeats. A basic combinatorial lemma is proved that will be used in the rest of the chapter. In the second section the Specht modules are defined. They are seen to give all the irreducible representations of S_n, and they have bases corresponding to standard tableaux with n boxes. The third section uses symmetric functions to prove some of the main theorems about these representations, including the character formula of Frobenius, Young's rule, and the branching formula. The last section contains a presentation of the Specht module as a quotient of a simpler representation; this will be useful in the next two chapters.

For brevity we assume a few of the basic facts about complex representations (always assumed to be finite-dimensional) of a finite group: that the number of irreducible representations, up to isomorphism, is the number of conjugacy classes; that the sum of the squares of the dimensions of these representations is the order of the group; that every representation decomposes into a sum of irreducible representations, each occurring with a certain multiplicity; that representations are determined by their characters. The orthogonality of characters, which is used to prove some of these facts, and the notion of induced representations, will also be assumed.