- 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
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