Editing Symmetric group (section)

== Representation theory ==
{{main article|Representation theory of the symmetric group}}

The [[representation theory of the symmetric group]] is a particular case of the [[representation theory of finite groups]], for which a concrete and detailed theory can be obtained. This has a large area of potential applications, from [[symmetric function]] theory to problems of [[quantum mechanics]] for a number of [[identical particles]].

The symmetric group S<sub>''n''</sub> has order ''n''<nowiki>!</nowiki>. Its [[conjugacy class]]es are labeled by [[integer partition|partition]]s of&nbsp;''n''. Therefore, according to the representation theory of a finite group, the number of inequivalent [[irreducible representation]]s, over the [[complex number]]s, is equal to the number of partitions of&nbsp;''n''. Unlike the general situation for finite groups, there is in fact a natural way to parametrize irreducible representation by the same set that parametrizes conjugacy classes, namely by partitions of ''n'' or equivalently [[Young diagram]]s of size&nbsp;''n''.

Each such irreducible representation can be realized over the integers (every permutation acting by a matrix with integer coefficients); it can be explicitly constructed by computing the [[Young symmetrizer]]s acting on a space generated by the [[Young tableau]]x of shape given by the Young diagram.

Over other [[Field (mathematics)|field]]s the situation can become much more complicated.  If the field ''K'' has [[characteristic (algebra)|characteristic]] equal to zero or greater than ''n'' then by [[Maschke's theorem]] the [[group ring|group algebra]] ''K''S<sub>''n''</sub> 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 [[module (mathematics)|module]]s 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. The modules so constructed are called ''[[Specht modules]]'', and every irreducible does arise inside some such module. There are now fewer irreducibles, and although they can be classified they are very poorly understood. For example, even their [[dimension (vector space)|dimension]]s are not known in general.

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.