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).