The Representations of the Symmetric Group

In this paper we classify the irreducible representations of the symmetric group Sn and give a proof of the hook formula for the dimension of each irreducible. 1. The Irreducible Representations of Sn We construct the irreducible representations of the symmetric group. The number of irreducible representations of Sn is the number of conjugacy classes of Sn, which is the number of partitions of n. Recall that a partition λ of n is λ = (λ1, . . . , λk) where n = λ1 + · · · + λk and λ1 ≥ · · · ≥ λk ≥ 1. To a partition λ = (λ1, . . . , λk) we associate a Young diagram with λi boxes in the ith row, the rows of boxes left-justified. We define a tableau on a given Young diagram to be a numbering of the boxes by the integers 1, 2, . . . , n, and we will call it standard if the rows and columns are increasing sequences. Below are examples of a Young diagram, a standard tableau, and a non-standard tableau, for n = 8, and λ = (3, 3, 2). 1 3 4 2 5 8 6 7 3 1 8 2 5 4 6 7 Young diagram standard tableau non-standard tableau