Representation theory of the symmetric group

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In mathematics, 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 Sn has order n!. 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. They are labeled by Young diagrams.

Each irreducible representation can be explicitly constructed from the Young diagrams by computing the Young symmetrizer for each tableau.

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 the group algebra KSn is semisimple. In these cases a construction of the irreducible representations can be carried out just as in the complex case.

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 construction of the irreducible modules over C can be replicated over any field, but the result will not in general be irreducible. The modules so constructed are called Specht modules, and every irreducible does arise in 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.

[edit] See also

[edit] References

  • William Fulton and Joe Harris, Representation Theory, A First Course (1991) Springer Verlag New York, ISBN 0-387-974495-4 See Chapter 4
  • 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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