What is Representation theory of the symmetric group?

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 quantum chemistry studies of atoms, molecules and solids.

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. 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 ; 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. The dimension d λ {\displaystyle d_{\lambda }} of the representation that corresponds to the Young diagram λ {\displaystyle \lambda } is given by the hook length formula.

To each irreducible representation ρ we can associate an irreducible character, χ.To compute χ where π is a permutation, one can use the combinatorial Murnaghan–Nakayama rule. Note that χ is constant on conjugacy classes,that is, χ = χ for all permutations σ.