# IrrRepSymNatPackage¶

irsn.spad line 1 [edit on github]

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).