Representations of Symmetric Groups — Mathematics Knowledge System

Overview

The irreducible complex representations of the symmetric group S_n are called Specht modules and are indexed by partitions of n (equivalently, by Young diagrams). Their characters — the characters of S_n — are given by the Murnaghan–Nakayama rule and the Frobenius character formula. The representation theory of symmetric groups is intimately linked to the theory of symmetric functions, Young tableaux, and combinatorics, making it one of the richest and most applicable corners of representation theory.

Intuition

Partition n as λ = (λ₁ ≥ λ₂ ≥ … ≥ λₖ > 0). Draw a Young diagram with λᵢ boxes in row i. The corresponding Specht module S^λ is an irreducible S_n-representation. Its dimension equals the number of standard Young tableaux of shape λ (cells filled with 1 through n, increasing along rows and columns) — this is given by the hook-length formula.

Formal Definition

Definition

Let λ ⊢ n be a partition of n. The Young symmetriser c_λ ∈ ℂ[S_n] is constructed from the row and column stabilisers of a standard Young tableau T of shape λ. The Specht module is S^λ = c_λ · ℂ[S_n].

\mathrm{Irr}(S_n) = \{S^\lambda : \lambda \vdash n\}

Irreducible representations indexed by partitions of n

\dim S^\lambda = f^\lambda = \frac{n!}{\prod_{(i,j) \in \lambda} h(i,j)}

Hook-length formula: f^λ = number of standard Young tableaux of shape λ

\sum_{\lambda \vdash n} (f^\lambda)^2 = n!

Sum of squares of dimensions equals n! = |S_n|

\chi^\lambda(\mu) = \frac{n!}{z_\mu}\langle s_\lambda, p_\mu \rangle

Frobenius character formula (s_λ = Schur function, p_μ = power-sum symmetric function, z_μ = centraliser size)

Notation

Notation	Meaning
$\lambda \vdash n$	Partition λ of n
$S^\lambda$	Specht module corresponding to partition λ
$f^\lambda$	Number of standard Young tableaux of shape λ (= dim S^λ)
$h(i,j)$	Hook length at cell (i,j) of Young diagram
$\chi^\lambda$	Character of Specht module S^λ

Properties

Character on cycle type

\chi^\lambda(\sigma) \text{ depends only on the cycle type of } \sigma

Dual/conjugate partition

S^{\lambda'} \cong S^\lambda \otimes \mathrm{sgn}_{S_n}, \text{ where } \lambda' \text{ is the conjugate partition}

Dimension of S^{(n)} and S^{(1,...,1)}

\dim S^{(n)} = 1 \;\text{(trivial rep)}, \quad \dim S^{(1^n)} = 1 \;\text{(sign rep)}

Theorems

Theorem 1: Classification of irreducible representations of S_n

The Specht modules \{S^\lambda : \lambda \vdash n\} form a complete set of pairwise non-isomorphic irreducible complex representations of S_n.

Theorem 2: Hook-length formula

\dim S^\lambda = f^\lambda = \frac{n!}{\prod_{(i,j) \in \lambda} h(i,j)}

Theorem 3: Schur–Weyl duality

(\mathbb{C}^m)^{\otimes n} \cong \bigoplus_{\lambda \vdash n,\; \ell(\lambda) \leq m} S^\lambda \otimes V^\lambda_m \quad \text{as } S_n \times GL_m(\mathbb{C})\text{-modules}

Worked Examples

1
The partitions of 3 are: (3), (2,1), (1,1,1).
$\lambda \vdash 3: (3),\; (2,1),\; (1,1,1)$
2
Dimensions: f^{(3)} = 1 (trivial), f^{(2,1)} = 2 (standard), f^{(1,1,1)} = 1 (sign).
$f^{(3)} = 1,\; f^{(2,1)} = 2,\; f^{(1,1,1)} = 1$
3
Check: 1² + 2² + 1² = 6 = |S₃|.
$1 + 4 + 1 = 6 = 3!$

✓ Answer

Three irreducibles corresponding to the three partitions of 3, with dimensions 1, 2, 1.

Practice Problems

Hardfree response

Find dim S^{(4,2,1)} using the hook-length formula.

Hardproof writing

State the Murnaghan–Nakayama rule and give an example of computing χ^{(3,1)}((1234)).

Common Mistakes

Common Mistake

Thinking Young tableaux and Young diagrams are the same thing

A Young diagram is a shape (the arrangement of boxes). A Young tableau is the same shape with numbers filled in. Standard Young tableaux have entries increasing along rows and columns; semistandard ones only require weakly increasing rows and strictly increasing columns.

Common Mistake

Assuming the representation theory of S_n is the same in all characteristics

Over ℂ (char 0), the Specht modules are all irreducible. Over a field of characteristic p, a Specht module S^λ may be reducible; its irreducible quotient is the James module D^λ. Modular representation theory of S_n is much harder.

Quiz

The irreducible complex representations of S_n are indexed by:

Subgroups of S_nConjugacy classes of S_nPartitions of nYoung tableaux of size n

The dimension of S^λ equals:

n!The number of standard Young tableaux of shape λThe number of conjugacy classes of S_nThe largest part of λ

Historical Background

Frobenius determined the irreducible characters of S_n in a celebrated 1900 paper using his newly developed character theory. Alfred Young's 1900–1902 papers introduced Young diagrams and Young tableaux independently. The algebraic construction of the Specht modules (via Young symmetrisers) was later made explicit and became the standard approach. Schur's dissertation (1901) related representations of S_n and GL_n via Schur–Weyl duality, a fundamental bridge between combinatorics and invariant theory.

1900
Frobenius determines irreducible characters of S_n using the Frobenius formula
Georg Frobenius
1900–1902
Alfred Young introduces Young diagrams and Young tableaux
Alfred Young
1901
Schur's dissertation establishes Schur–Weyl duality
Issai Schur
1977
James's book 'The Representation Theory of the Symmetric Groups' modernises the Specht module approach
Gordon James

Summary

The irreducible complex representations of S_n (Specht modules S^λ) are indexed by partitions λ ⊢ n.
dim S^λ equals the number of standard Young tableaux of shape λ, given by the hook-length formula.
Characters are computed via the Murnaghan–Nakayama rule or Frobenius character formula.
Schur–Weyl duality relates irreps of S_n and GL_m via their joint action on (ℂ^m)^{⊗n}.
Over fields of positive characteristic, the Specht modules may be reducible (modular representation theory).

References

BookSagan, B. — The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, 2nd ed. (2001)
BookJames, G. — The Representation Theory of the Symmetric Groups (1978)
BookFulton, W. — Young Tableaux (1997)
WebsiteWikipedia — Representation theory of the symmetric group
WebsiteMathWorld — Symmetric Group Representation