Representation Theory of The Symmetric Group - Low Dimensional Representations

Low Dimensional Representations

The lowest dimensional representations of the symmetric groups can be described explicitly, as done in (Burnside 1955, p. 468). This work was extended to the smallest k degrees (explicitly for k=4, and k=7) in (Rasala 1977), and over arbitrary fields in (James 1983). The smallest two degrees in characteristic zero are described here:

Every symmetric group has a one-dimensional representation called the trivial representation, where every element acts as the one by one identity matrix. For n ≥ 2, there is another irreducible representation of degree 1, called the sign representation or alternating character, which takes a permutation to the one by one matrix with entry ±1 based on the sign of the permutation. These are the only one-dimensional representations of the symmetric groups, as one-dimensional representations are abelian, and the abelianization of the symmetric group is the cyclic group of order 2.

For all n, there is an n-dimensional representation of the symmetric group of order n, called the permutation representation, which consists of permuting n coordinates. This has the trivial subrepresentation consisting of vectors whose coordinates are all equal. The orthogonal complement consists of those vectors whose coordinates sum to zero, and when n ≥ 2, the representation on this subspace is an n—1 dimensional irreducible representation. Another n−1 dimensional irreducible representation is found by tensoring with the sign representation.

For n ≥ 7, these are the lowest-dimensional irreducible representations of S_n – all other irreducible representations have dimension at least n. However for n = 4, the surjection from S₄ to S₃ allows S₄ to inherit a two-dimensional irreducible representation. For n = 6, the exceptional transitive embedding of S₅ into S₆ produces another pair of five-dimensional irreducible representations.