# Representation theory of the symmetric group

In

mathematics , the**representation theory of the symmetric group**is a particular case of therepresentation theory of finite groups , for which a concrete and detailed theory can be obtained. This has a large area of potential applications, fromsymmetric function theory to problems ofquantum mechanics for a number ofidentical particles .The

symmetric group "S"_{"n"}has order "n"!. Itsconjugacy class es are labeled by partitions of "n". Therefore according to the representation theory of a finite group, the number of inequivalentirreducible representation s, over thecomplex number s, 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 representation by the same set that parametrizes conjugacy classes, namely by partitions of "n" or equivalentlyYoung diagram s 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 symmetrizer s acting on a space generated by theYoung tableau x of shape given by the Young diagram.Over other fields the situation can become much more complicated. If the field "K" has characteristic equal to zero or greater than "n" then by

Maschke's theorem thegroup algebra "KS"_{"n"}is semisimple. In these cases the irreducible representations defined over the integers give the complete set of irreducible representations (after reduction modulo the characteristic is 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. 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 dimensions 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.

**ee also***

Robinson-Schensted algorithm

*Schur–Weyl duality **References*** Lecture 4 of Fulton-Harris

* Gordon James and Adalbert Kerber, "The representation theory of the symmetric group" (1984) Cambridge University Press, ISBN 0-521-30236-6

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Symmetric group**— Not to be confused with Symmetry group. A Cayley graph of the symmetric group S4 … Wikipedia**Representation theory of finite groups**— In mathematics, representation theory is a technique for analyzing abstract groups in terms of groups of linear transformations. See the article on group representations for an introduction. This article discusses the representation theory of… … Wikipedia**Representation theory**— This article is about the theory of representations of algebraic structures by linear transformations and matrices. For the more general notion of representations throughout mathematics, see representation (mathematics). Representation theory is… … Wikipedia**List of representation theory topics**— This is a list of representation theory topics, by Wikipedia page. See also list of harmonic analysis topics, which is more directed towards the mathematical analysis aspects of representation theory. Contents 1 General representation theory 2… … Wikipedia**Modular representation theory**— is a branch of mathematics, and that part of representation theory that studies linear representations of finite group G over a field K of positive characteristic. As well as having applications to group theory, modular representations arise… … Wikipedia**Symmetric algebra**— In mathematics, the symmetric algebra S ( V ) (also denoted Sym ( V )) on a vector space V over a field K is the free commutative unital associative K algebra containing V .It corresponds to polynomials with indeterminates in V , without choosing … Wikipedia**Symmetric function**— In mathematics, the term symmetric function can mean two different concepts. A symmetric function of n variables is one whose value at any n tuple of arguments is the same as its value at any permutation of that n tuple. While this notion can… … Wikipedia**Group theory**— is a mathematical discipline, the part of abstract algebra that studies the algebraic structures known as groups. The development of group theory sprang from three main sources: number theory, theory of algebraic equations, and geometry. The… … Wikipedia**Group (mathematics)**— This article covers basic notions. For advanced topics, see Group theory. The possible manipulations of this Rubik s Cube form a group. In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines … Wikipedia**Group action**— This article is about the mathematical concept. For the sociology term, see group action (sociology). Given an equilateral triangle, the counterclockwise rotation by 120° around the center of the triangle acts on the set of vertices of the… … Wikipedia