Symmetric groups

The lowest-dimensional representations of the symmetric groups can be described explicitly,^[4][5] and over arbitrary fields.^[6] 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 C₂, the cyclic group of order 2.

For all n, there is an n-dimensional representation of the symmetric group of order n!, called the natural 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, called the standard representation. Another (n − 1)-dimensional irreducible representation is found by tensoring with the sign representation. An exterior power ${\displaystyle \Lambda ^{k}V}$ of the standard representation ${\displaystyle V}$ is irreducible provided ${\displaystyle 0\leq k\leq n-1}$ (Fulton & Harris 2004).

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.

Irreducible representation of ${\displaystyle S_{n}}$	Dimension	Young diagram of size ${\displaystyle n}$
Trivial representation	${\displaystyle 1}$	${\displaystyle (n)}$
Sign representation	${\displaystyle 1}$	${\displaystyle (1^{n})=(1,1,\dots ,1)}$
Standard representation ${\displaystyle V}$	${\displaystyle n-1}$	${\displaystyle (n-1,1)}$
Exterior power ${\displaystyle \Lambda ^{k}V}$	${\displaystyle {\binom {n-1}{k}}}$	${\displaystyle (n-k,1^{k})}$

Alternating groups

The compound of five tetrahedra, on which A₅ acts, giving a 3-dimensional representation.

The representation theory of the alternating groups is similar, though the sign representation disappears. For n ≥ 7, the lowest-dimensional irreducible representations are the trivial representation in dimension one, and the (n − 1)-dimensional representation from the other summand of the permutation representation, with all other irreducible representations having higher dimension, but there are exceptions for smaller n.

The alternating groups for n ≥ 5 have only one one-dimensional irreducible representation, the trivial representation. For n = 3, 4 there are two additional one-dimensional irreducible representations, corresponding to maps to the cyclic group of order 3: A₃ ≅ C₃ and A₄ → A₄/V ≅ C₃.

• For n ≥ 7, there is just one irreducible representation of degree n − 1, and this is the smallest degree of a non-trivial irreducible representation.
• For n = 3 the obvious analogue of the (n − 1)-dimensional representation is reducible – the permutation representation coincides with the regular representation, and thus breaks up into the three one-dimensional representations, as A₃ ≅ C₃ is abelian; see the discrete Fourier transform for representation theory of cyclic groups.
• For n = 4, there is just one n − 1 irreducible representation, but there are the exceptional irreducible representations of dimension 1.
• For n = 5, there are two dual irreducible representations of dimension 3, corresponding to its action as icosahedral symmetry.
• For n = 6, there is an extra irreducible representation of dimension 5 corresponding to the exceptional transitive embedding of A₅ in A₆.

Kronecker coefficients

The tensor product of two representations of ${\displaystyle S_{n}}$ corresponding to the Young diagrams ${\displaystyle \lambda ,\mu }$ is a combination of irreducible representations of ${\displaystyle S_{n}}$,

${\displaystyle V_{\lambda }\otimes V_{\mu }\cong \sum _{\nu }C_{\lambda ,\mu ,\nu }V_{\nu }}$

The coefficients ${\displaystyle C_{\lambda \mu \nu }\in \mathbb {N} }$ are called the Kronecker coefficients of the symmetric group. They can be computed from the characters of the representations (Fulton & Harris 2004):

${\displaystyle C_{\lambda ,\mu ,\nu }=\sum _{\rho }{\frac {1}{z_{\rho }}}\chi _{\lambda }(C_{\rho })\chi _{\mu }(C_{\rho })\chi _{\nu }(C_{\rho })}$

The sum is over partitions ${\displaystyle \rho }$ of ${\displaystyle n}$, with ${\displaystyle C_{\rho }}$ the corresponding conjugacy classes. The values of the characters ${\displaystyle \chi _{\lambda }(C_{\rho })}$ can be computed using the Frobenius formula. The coefficients ${\displaystyle z_{\rho }}$ are

${\displaystyle z_{\rho }=\prod _{j=0}^{n}j^{i_{j}}i_{j}!={\frac {n!}{|C_{\rho }|}}}$

where ${\displaystyle i_{j}}$ is the number of times ${\displaystyle j}$ appears in ${\displaystyle \rho }$, so that ${\displaystyle \sum i_{j}j=n}$.

A few examples, written in terms of Young diagrams (Hamermesh 1989):

${\displaystyle (n-1,1)\otimes (n-1,1)\cong (n)+(n-1,1)+(n-2,2)+(n-2,1,1)}$

${\displaystyle (n-1,1)\otimes (n-2,2){\underset {n>4}{\cong }}(n-1,1)+(n-2,2)+(n-2,1,1)+(n-3,3)+(n-3,2,1)}$

${\displaystyle (n-1,1)\otimes (n-2,1,1)\cong (n-1,1)+(n-2,2)+(n-2,1,1)+(n-3,2,1)+(n-3,1,1,1)}$

${\displaystyle {\begin{aligned}(n-2,2)\otimes (n-2,2)\cong &(n)+(n-1,1)+2(n-2,2)+(n-2,1,1)+(n-3,3)\\&+2(n-3,2,1)+(n-3,1,1,1)+(n-4,4)+(n-4,3,1)+(n-4,2,2)\end{aligned}}}$

There is a simple rule for computing ${\displaystyle (n-1,1)\otimes \lambda }$ for any Young diagram ${\displaystyle \lambda }$ (Hamermesh 1989): the result is the sum of all Young diagrams that are obtained from ${\displaystyle \lambda }$ by removing one box and then adding one box, where the coefficients are one except for ${\displaystyle \lambda }$ itself, whose coefficient is ${\displaystyle \#\{\lambda _{i}\}-1}$, i.e., the number of different row lengths minus one.

A constraint on the irreducible constituents of ${\displaystyle V_{\lambda }\otimes V_{\mu }}$ is (James & Kerber 1981)

${\displaystyle C_{\lambda ,\mu ,\nu }>0\implies |d_{\lambda }-d_{\mu }|\leq d_{\nu }\leq d_{\lambda }+d_{\mu }}$

where the depth ${\displaystyle d_{\lambda }=n-\lambda _{1}}$ of a Young diagram is the number of boxes that do not belong to the first row.

Reduced Kronecker coefficients

For ${\displaystyle \lambda }$ a Young diagram and ${\displaystyle n\geq \lambda _{1}}$, ${\displaystyle \lambda [n]=(n-|\lambda |,\lambda )}$ is a Young diagram of size ${\displaystyle n}$. Then ${\displaystyle C_{\lambda [n],\mu [n],\nu [n]}}$ is a bounded, non-decreasing function of ${\displaystyle n}$, and

${\displaystyle {\bar {C}}_{\lambda ,\mu ,\nu }=\lim _{n\to \infty }C_{\lambda [n],\mu [n],\nu [n]}}$

is called a reduced Kronecker coefficient^[7] or stable Kronecker coefficient.^[8] There are known bounds on the value of ${\displaystyle n}$ where ${\displaystyle C_{\lambda [n],\mu [n],\nu [n]}}$ reaches its limit.^[7] The reduced Kronecker coefficients are structure constants of Deligne categories of representations of ${\displaystyle S_{n}}$ with ${\displaystyle n\in \mathbb {C} -\mathbb {N} }$.^[9]

In contrast to Kronecker coefficients, reduced Kronecker coefficients are defined for any triple of Young diagrams, not necessarily of the same size. If ${\displaystyle |\nu |=|\lambda |+|\mu |}$, then ${\displaystyle {\bar {C}}_{\lambda ,\mu ,\nu }}$ coincides with the Littlewood-Richardson coefficient ${\displaystyle c_{\lambda ,\mu }^{\nu }}$.^[10] Reduced Kronecker coefficients can be written as linear combinations of Littlewood-Richardson coefficients via a change of bases in the space of symmetric functions, giving rise to expressions that are manifestly integral although not manifestly positive.^[8] Reduced Kronecker coefficients can also be written in terms of Kronecker and Littlewood-Richardson coefficients ${\displaystyle c_{\alpha \beta \gamma }^{\lambda }}$ via Littlewood's formula^[11][12]

${\displaystyle {\bar {C}}_{\lambda ,\mu ,\nu }=\sum _{\lambda ',\mu ',\nu ',\alpha ,\beta ,\gamma }C_{\lambda ',\mu ',\nu '}c_{\lambda '\beta \gamma }^{\lambda }c_{\mu '\alpha \gamma }^{\mu }c_{\nu '\alpha \beta }^{\nu }}$

Conversely, it is possible to recover the Kronecker coefficients as linear combinations of reduced Kronecker coefficients.^[7]

Reduced Kronecker coefficients are implemented in the computer algebra system SageMath.^[13][14]