REPRESENTATIONS OF Sn AND GL(n,C)

For a given finite group G, we have that the number of irreducible representations of G is equal to the number of conjugacy classes of G. Although numerically equal, there is no general formula to map the conjugacy classes of a group to its irreducible representations. For the case of the symmetric group Sn, however, there is a remarkably simple correspondence; we will see that each irreducible representation of Sn is determined uniquely by a combinatorial object called a Young diagram corresponding to a given conjugacy class. In section 2, we will introduce these diagrams and show how they determine a representation Vλ of Sn. In section 3, we will show that each such representation is irreducible and, in fact, the collection of Vλ’s accounts for every irreducible representation of Sn. Finally, in section 4, we will discuss how these representations can be used to give a large class of irreducible representations of the general linear group GLk. For more details, please refer to Chapters 4, 6, and 15 of Representation Theory: A First Course by William Fulton and Joe Harris.