# IrrRepSymNatPackage¶

IrrRepSymNatPackage contains functions for computing the ordinary irreducible representations of symmetric groups on `n` letters {1, 2, …, n} in Young`'s` natural form and their dimensions. These representations can be labelled by number partitions of `n`, i.e. a weakly decreasing sequence of integers summing up to `n`, e.g. [3, 3, 3, 1] labels an irreducible representation for `n` equals 10. Note: whenever a List Integer appears in a signature, a partition required.

dimensionOfIrreducibleRepresentation: List Integer -> NonNegativeInteger

`dimensionOfIrreducibleRepresentation(lambda)` is the dimension of the ordinary irreducible representation of the symmetric group corresponding to lambda. Note: the Robinson-Thrall hook formula is implemented.

irreducibleRepresentation: (List Integer, List Permutation Integer) -> List Matrix Integer

`irreducibleRepresentation(lambda, listOfPerm)` is the list of the irreducible representations corresponding to lambda in Young`'s` natural form for the list of permutations given by listOfPerm.

irreducibleRepresentation: (List Integer, Permutation Integer) -> Matrix Integer

`irreducibleRepresentation(lambda, pi)` is the irreducible representation corresponding to partition lambda in Young`'s` natural form of the permutation ``pi``* in the symmetric group, whose elements permute *{1, 2, …, n}.

irreducibleRepresentation: List Integer -> List Matrix Integer

`irreducibleRepresentation(lambda)` is the list of the two irreducible representations corresponding to the partition lambda in Young`'s` natural form for the following two generators of the symmetric group, whose elements permute {1, 2, …, n}, namely (1 2) (2-cycle) and (1 2 … n) (`n`-cycle).