Representation theory of the symmetric group

This has a large area of potential applications, from symmetric function theory to quantum chemistry studies of atoms, molecules and solids.

Its conjugacy classes are labeled by partitions of n. Therefore according to the representation theory of a finite group, the number of inequivalent irreducible representations, over the complex numbers, is equal to the number of partitions of n. Unlike the general situation for finite groups, there is in fact a natural way to parametrize irreducible representations by the same set that parametrizes conjugacy classes, namely by partitions of n or equivalently Young diagrams of size n. Each such irreducible representation can in fact be realized over the integers (every permutation acting by a matrix with integer coefficients); it can be explicitly constructed by computing the Young symmetrizers acting on a space generated by the Young tableaux of shape given by the Young diagram.

If the field K has characteristic equal to zero or greater than n then by Maschke's theorem the group algebra KSn is semisimple.

In these cases the irreducible representations defined over the integers give the complete set of irreducible representations (after reduction modulo the characteristic if necessary).

However, the irreducible representations of the symmetric group are not known in arbitrary characteristic.

In this context it is more usual to use the language of modules rather than representations.

The representation obtained from an irreducible representation defined over the integers by reducing modulo the characteristic will not in general be irreducible.

There are now fewer irreducibles, and although they can be classified they are very poorly understood.

The determination of the irreducible modules for the symmetric group over an arbitrary field is widely regarded as one of the most important open problems in representation theory.

The lowest-dimensional representations of the symmetric groups can be described explicitly,[4][5] and over arbitrary fields.

[6][page needed] 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.

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.

For n = 6, the exceptional transitive embedding of S5 into S6 produces another pair of five-dimensional irreducible representations.

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.

are called the Kronecker coefficients of the symmetric group.

They can be computed from the characters of the representations (Fulton & Harris 2004): The sum is over partitions

A few examples, written in terms of Young diagrams (Hamermesh 1989): There is a simple rule for computing

(Hamermesh 1989): the result is the sum of all Young diagrams that are obtained from

, i.e., the number of different row lengths minus one.

of a Young diagram is the number of boxes that do not belong to the first row.

[7] The reduced Kronecker coefficients are structure constants of Deligne categories of representations of

[9] In contrast to Kronecker coefficients, reduced Kronecker coefficients are defined for any triple of Young diagrams, not necessarily of the same size.

[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.

via Littlewood's formula[11][12] 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.

[15] There is a combinatorial description of the cyclic exponents of the symmetric group (and wreath products thereof).

-index of a standard Young tableau be the sum of the values of

then describe how it decomposes into representations of the cyclic group