Representations of Symmetric Groups

Core Idea

The irreducible representations of the symmetric group Sₙ over ℂ are indexed by partitions of n. Each partition λ ⊢ n gives an irreducible representation Sλ (the Specht module) whose dimension equals the number of standard Young tableaux of shape λ. This parametrization connects representation theory to combinatorics: the deep structure of Sₙ-representations is encoded in the combinatorics of partitions, tableaux, and symmetric functions.

Explainer

The symmetric group Sₙ — the group of all permutations of {1, …, n} — is one of the most important groups in mathematics, and its representation theory is correspondingly rich. The key structural fact is that conjugacy classes in Sₙ are determined by cycle type: two permutations are conjugate if and only if they have the same partition into disjoint cycles. Since cycle types are exactly partitions of n, the number of conjugacy classes (and hence irreducible representations) equals p(n), the number of partitions.

The irreducible representations are the Specht modules Sλ, one for each partition λ ⊢ n. The construction uses Young tableaux: fill the Young diagram of λ with the numbers 1, …, n to get a Young tableau, then use symmetrization and antisymmetrization operations on the rows and columns to build an irreducible subspace of the regular representation. The dimension of Sλ equals the number of standard Young tableaux of shape λ (fillings where entries increase along rows and down columns), computed by the elegant hook length formula: dim(Sλ) = n! / ∏ h(□).

For S₃, the partitions of 3 are (3), (2,1), (1,1,1). The partition (3) gives the trivial representation (dimension 1). The partition (1,1,1) gives the sign representation (dimension 1). The partition (2,1) gives a 2-dimensional representation — the standard representation, where S₃ acts on the plane {(x₁,x₂,x₃) : x₁+x₂+x₃ = 0} by permuting coordinates. The dimensions check: 1² + 2² + 1² = 6 = 3!.

The representation theory of Sₙ connects to a vast web of mathematics. The characters of Sₙ are given by symmetric functions (Schur functions), linking to algebraic combinatorics. The branching rules (how Sₙ-representations restrict to Sₙ₋₁) are governed by removing boxes from Young diagrams, connecting to the theory of symmetric functions and the RSK correspondence. Through Schur-Weyl duality, the representations of Sₙ are intimately related to the representations of GL_n — the combinatorics of partitions serves both.