Linear representation theory of symmetric groups

This article discusses the irreducible linear representations of the symmetric groups of finite degree, i.e., the group of all permutations on a finite set. We denote by $$S_n$$ the symmetric group of degree $$n$$. For convenience, we assume that the set it acts on is $$\{ 1,2,\dots,n \}$$.

All symmetric groups are rational-representation groups: all their representations can be realized over the field of rational numbers. In particular, all the character values are (rational) integers, i.e., elements of $$\mathbb{Z}$$. In fact, the representations themselves can be realized using matrices where all the entries are integers. (Note that the representations being realizable over $$\mathbb{Q}$$ is equivalent to their being realizable over $$\mathbb{Z}$$ because linear representation is realizable over principal ideal domain iff it is realizable over field of fractions).

We discuss here how to parametrize the irreducible representations and compute the character values.

It turns out that the set of irreducible representations is parametrized by the set of unordered integer partitions of $$n$$. Because cycle type determines conjugacy class, the unordered integer partitions also parametrize the conjugacy classes via the cycle type map. Thus, the symmetric groups of degree $$n$$ come with a bijection between conjugacy classes and irreducible representations, something that does not generalize to all finite groups even though the number of conjugacy classes equals the number of irreducible representations for all finite groups.

Hook-length formula
From the above, it is clear that the degree of the irreducible representation corresponding to an unordered integer partition $$\lambda$$ equals the number of Young tableaux whose Young diagram has shape $$\lambda$$. This number is given by the hook-length formula:

$$d_\lambda = \frac{n!}{\prod \operatorname{hook-lengths}}$$.

Here, the hook length of a position in a Young diagram is the total number of positions that are directly below it, directly to the right of it, or equal to it. For instance, for the partition $$3 + 2 + 1$$, the hook lengths of corner positions are all $$1$$, the hook length of the top left square is $$5$$, and the hook lengths of the other two squares are both $$3$$. Thus, the degree of the corresponding irreducible representation is:

$$\frac{6!}{5 \cdot 3 \cdot 3} = 16$$.

Note that this formula confirms several things we already know:


 * Degree of irreducible representation divides order of group
 * Sum of squares of degrees of irreducible representations equals order of group (this follows from the above and the Robinson-Schensted correspondence, which puts the symmetric group's elements in bijection with pairs of tableaux of the same shape.

Inductive relationships
The degree of the irreducible representation corresponding to a partition $$\lambda$$ is equal to the number of Young tableaux of shape $$\lambda$$. A Young tableau of shape $$\lambda$$ can be identified with a directed path in the Young graph from the one-element Young diagram to the diagram for shape $$\lambda$$. The path basically describes the order in which boxes are added, which in turn can be encoded by numbering the squares, giving a Young tableau.

From this, it is easy to see the following:


 * The degree of the irreducible representation corresponding to $$\lambda$$ equals the number of paths from the one-box Young diagram (i.e., the trivial partition of $$1$$) to $$\lambda$$.
 * The degree of the irreducible representation corresponding to $$\lambda$$ is the sum of the degrees of irreducible representations corresponding to all partitions obtained by removing one box from $$\lambda$$. In other words, in the Young graph, the degree of the irreducible representation corresponding to a partition is the sum of the degrees of irreducible representations corresponding to its parents.
 * $$n$$ times the degree of the irreducible representation corresponding to a partition $$\lambda$$ of $$n - 1$$ is the sum of the degrees of irreducible representations of the children of $$\lambda$$.

These observations are also related to the notions of restriction/induction of representations, as discussed later in the article.

Listing of degrees of irreducible representations for symmetric groups of small degree
The graph here shows partitions and the corresponding degrees of irreducible representations, where an arrow from on partition to another means that the Young diagram of the latter can be obtained by adding one box to the Young diagram of the former. The full graph, which is infinite, is termed the Young graph.



More comprehensive listing of degrees
Below is a comprehensive listing of degrees of irreducible representations of $$S_n$$ for $$1 \le n \le 9$$, along with some related information. The Plancherel measure of a representation of degree $$d$$ is $$d^2/(n!)$$:

See also the later section on asymptotic representation theory.

Special cases of hook partitions
The two linear Young diagrams (all vertical and all horizontal) correspond to the two one-dimensional representations: the trivial representation and the sign representation.

The partitions whose shapes are hooks, and are of the form $$a + 1 + 1 + \dots + 1$$ correspond to important representations. The degree of the representation for such a partition is $$\binom{n - 1}{a - 1}$$. Particular cases of importance are:


 * $$a = n$$, i.e., the horizontal diagram. This corresponds to the trivial representation. It has degree $$1$$, and sends every permutation to $$(1)$$.
 * $$a = 1$$, i.e., the vertical diagram. This corresponds to the sign representation. It has degree $$-1$$, and sends every permutation to its sign: $$(1)$$ for even permutations and $$(-1)nb$$ for odd permutations.
 * $$a = n - 1$$. This corresponds to the standard representation. This is the nontrivial irreducible component of the representation of the symmetric group on a $$n$$-dimensional vector space by permutation on the coordinates. It has degree $$n - 1$$. The corresponding character sends each permutation to one less than its number of fixed points.
 * $$a = 2$$: This corresponds to the tensor product of the standard representation and the sign representation. It has degree $$n - 1$$. The corresponding character sends every permutation to one less than its number of fixed points if the permutation is an even permutation and the negative of (one less than the number of fixed points) if the permutation is an odd permutation.

Conjugate of a partition
For any partition $$\lambda$$, there is a conjugate partition $$\overline{\lambda}$$ obtained geometrically by flipping the Young diagram (interchanging the role of rows and columns). The hook-length formula makes it clear that the degrees $$d_\lambda$$ and $$d_{\overline{\lambda}}$$ are equal. Further, the description of the construction of the irreducible representation makes it clear that the representation corresponding to $$\overline{\lambda}$$ is equivalent to the representation obtained by multiplying the representation corresponding to $$\lambda$$ by multiplying by the sign of the permutation. In other words, the representation for $$\overline{\lambda}$$ is the tensor product of the sign representation and the representation for $$\lambda$$. Thus, the character values for these representations are equal for even permutations and negatives of each other for odd permutations.

A partition $$\lambda$$ that equals its own conjugate partition is termed a self-conjugate partition. Examples are $$2 + 1, 3 + 2 + 1, 4 + 2 + 1 + 1$$. In particular, the representation corresponding to a self-conjugate partition is equivalent to its tensor product with the sign representation. Thus, the character values of the representation corresponding to a self-conjugate partition must be $$0$$ at all odd permutations.

Case of one less
Consider the embedding of $$S_{n - 1}$$ in $$S_n$$ induced by the inclusion map $$\{ 1,2,3, \dots, n- 1\}$$ to $$\{1 ,2,3 \dots, n \}$$. Suppose $$\varphi$$ is an irreducible representation of $$S_{n-1}$$ and $$\psi$$ is an irreducible representation of $$S_n$$.

By Frobenius reciprocity, the multiplicity of $$\varphi$$ in the restriction of $$\psi$$ to $$S_{n-1}$$ equals the multiplicity of $$\psi$$ in the induction of $$\varphi$$ to $$S_n$$. It turns out that the following is true as a special case of Pieri's rule:


 * If the Young diagram corresponding to $$\psi$$ can be obtained by adding one box to the Young diagram corresponding to $$\varphi$$, then both these multiplicites are one. This is equivalent to saying that in the Young graph, there is an edge directed from $$\varphi$$ to $$\psi$$.
 * Otherwise, both multiplicities are zero.

Some easy corollaries of this:


 * The degree of the irreducible representation corresponding to a partition of $$n$$ is the sum of the degrees of all irreducible representations corresponding to partitions of $$n - 1$$ that are its parents in the Young graph, i.e., such that adding one box to the Young diagram gives the partition of $$n$$
 * $$n$$ times the degree of an irreducible representation corresponding to a partition of $$n - 1$$ is the sum of degrees of all irreducible representations corresponding to its children in the Young graph, i.e., all Young diagrams that can be obtained by adding one box.

These are easily illustrated in the Young graph:



Character tables
Given below are the character tables for small values of $$n$$ as matrices. The representations have been arranged in increasing order of degree.

Asymptotic representation theory
Everywhere below, $$n$$ denotes the degree of the symmetric group under consideration, and we are interested in the asymptotic behavior as $$n \to \infty$$:

Character table
The character table can be constructed using GAP's CharacterTable function. The character table for the symmetric group of degree $$n$$ can be constructed using either of the following functions:

CharacterTable(SymmetricGroup(n))

or

CharacterTable("Symmetric",n)

The characters of irreducible representations can be accessed using the Irr function, with each entry recording the values of a character taken on various conjugacy classes. For instance, here is a determination of the character table of the symmetric group of degree 8:

gap> C := CharacterTable("Symmetric",8); CharacterTable( "Sym(8)" ) gap> Irr(C); [ Character( CharacterTable( "Sym(8)" ), [ 1, -1, 1, -1, 1, 1, -1, 1, 1, -1, -1, 1, -1, -1, 1, 1, -1, 1, -1, 1, 1, -1 ] ), Character( CharacterTable( "Sym(8)" ), [ 7, -5, 3, -1, -1, 4, -2, 0, 1, 1, -3, 1, 1, 0, -1, 2, 0, -1, -1, -1, 0, 1 ] ), Character( CharacterTable( "Sym(8)" ), [ 20, -10, 4, -2, 4, 5, -1, 1, -1, -1, -2, 0, -2, 1, 0, 0, 0, 0, 1, 1, -1, 0 ] ), Character( CharacterTable( "Sym(8)" ), [ 28, -10, 4, -2, -4, 1, -1, 1, 1, -1, 2, 0, 2, -1, 0, -2, 0, 1, 1, -1, 0, 0 ] ), Character( CharacterTable( "Sym(8)" ), [ 14, -4, 2, 0, 6, -1, -1, -1, 2, 2, 2, 0, -2, -1, 2, -1, 1, -1, 0, 0, 0, 0 ] ), Character( CharacterTable( "Sym(8)" ), [ 21, -9, 1, 3, -3, 6, 0, -2, 0, 0, -3, -1, 1, 0, 1, 1, 1, 1, 0, 0, 0, -1 ] ), Character( CharacterTable( "Sym(8)" ), [ 64, -16, 0, 0, 0, 4, 2, 0, -2, 2, 0, 0, 0, 0, 0, -1, -1, -1, 0, 0, 1, 0 ] ), Character( CharacterTable( "Sym(8)" ), [ 70, -10, 2, 2, -2, -5, -1, -1, 1, -1, 4, 0, 0, 1, -2, 0, 0, 0, -1, 1, 0, 0 ] ), Character( CharacterTable( "Sym(8)" ), [ 56, -4, 0, -4, 8, -4, 2, 0, -1, -1, 0, 0, 0, 0, 0, 1, 1, 1, -1, -1, 0, 0 ] ), Character( CharacterTable( "Sym(8)" ), [ 42, 0, 2, 0, -6, -6, 0, 2, 0, 0, 0, -2, 0, 0, 2, 2, 0, -1, 0, 0, 0, 0 ] ), Character( CharacterTable( "Sym(8)" ), [ 35, -5, -5, 3, 3, 5, 1, 1, 2, -2, -1, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 1 ] ), Character( CharacterTable( "Sym(8)" ), [ 90, 0, -6, 0, -6, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, -1, 0 ] ), Character( CharacterTable( "Sym(8)" ), [ 56, 4, 0, 4, 8, -4, -2, 0, -1, 1, 0, 0, 0, 0, 0, 1, -1, 1, 1, -1, 0, 0 ] ), Character( CharacterTable( "Sym(8)" ), [ 70, 10, 2, -2, -2, -5, 1, -1, 1, 1, -4, 0, 0, -1, -2, 0, 0, 0, 1, 1, 0, 0 ] ), Character( CharacterTable( "Sym(8)" ), [ 14, 4, 2, 0, 6, -1, 1, -1, 2, -2, -2, 0, 2, 1, 2, -1, -1, -1, 0, 0, 0, 0 ] ), Character( CharacterTable( "Sym(8)" ), [ 35, 5, -5, -3, 3, 5, -1, 1, 2, 2, 1, -1, 1, 1, -1, 0, 0, 0, 0, 0, 0, -1 ] ), Character( CharacterTable( "Sym(8)" ), [ 64, 16, 0, 0, 0, 4, -2, 0, -2, -2, 0, 0, 0, 0, 0, -1, 1, -1, 0, 0, 1, 0 ] ), Character( CharacterTable( "Sym(8)" ), [ 28, 10, 4, 2, -4, 1, 1, 1, 1, 1, -2, 0, -2, 1, 0, -2, 0, 1, -1, -1, 0, 0 ] ), Character( CharacterTable( "Sym(8)" ), [ 21, 9, 1, -3, -3, 6, 0, -2, 0, 0, 3, -1, -1, 0, 1, 1, -1, 1, 0, 0, 0, 1 ] ), Character( CharacterTable( "Sym(8)" ), [ 20, 10, 4, 2, 4, 5, 1, 1, -1, 1, 2, 0, 2, -1, 0, 0, 0, 0, -1, 1, -1, 0 ] ), Character( CharacterTable( "Sym(8)" ), [ 7, 5, 3, 1, -1, 4, 2, 0, 1, -1, 3, 1, -1, 0, -1, 2, 0, -1, 1, -1, 0, -1 ] ), Character( CharacterTable( "Sym(8)" ), [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ) ]

Irreducible representations
Explicit matrices for the irreducible representations on a generating set can be obtained using GAP's IrreducibleRepresentations function. For instance, for $$n = 4$$, this looks like:

gap> IrreducibleRepresentations(SymmetricGroup(4)); [ Pcgs([ (3,4), (2,4,3), (1,4)(2,3), (1,3)(2,4) ]) -> [ [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ (3,4), (2,4,3), (1,4)(2,3), (1,3)(2,4) ]) -> [ [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ (3,4), (2,4,3), (1,4)(2,3), (1,3)(2,4) ]) -> [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ E(3), 0 ], [ 0, E(3)^2 ] ], [ [ 1, 0 ], [ 0, 1 ] ], [ [ 1, 0 ], [ 0, 1 ] ] ], Pcgs([ (3,4), (2,4,3), (1,4)(2,3), (1,3)(2,4) ]) -> [ [ [ 0, 1, 0 ], [ 1, 0, 0 ], [ 0, 0, 1 ] ], [ [ 0, 0, 1 ], [ 1, 0, 0 ], [ 0, 1, 0 ] ], [ [ -1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, -1 ] ], [ [ 1, 0, 0 ], [ 0, -1, 0 ], [ 0, 0, -1 ] ] ], Pcgs([ (3,4), (2,4,3), (1,4)(2,3), (1,3)(2,4) ]) -> [ [ [ 0, -1, 0 ], [ -1, 0, 0 ], [ 0, 0, -1 ] ], [ [ 0, 0, 1 ], [ 1, 0, 0 ], [ 0, 1, 0 ] ], [ [ -1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, -1 ] ],     [ [ 1, 0, 0 ], [ 0, -1, 0 ], [ 0, 0, -1 ] ] ] ]

Note that the matrices are specified only on a generating set and not on all elements of the symmetric group.