Labels for irreducible symmetric group characters

Question

Using GAP I can produce irreducible characters for symmetric group easily enough. For example:

G:=SymmetricGroup(5);
Irr(G);

Then, for example Irr(G)[4]; outputs:

Character( CharacterTable( Sym( [ 1 .. 5 ] ) ), [ 6, 0, -2, 0, 0, 0, 1 ] )

However the irreducible characters are labelled by partitions. Is there a way I can get GAP to tell me that Irr(G)[4], for example, is labelled by the partition [3,1,1].

Thank you.

Answer 1

If you first create the group, and then calculate the character table, this is done by a generic character table routine that does not know the special representation theory of the symmetric group.

There is an alternative way to calculate the character table of a symmetric group, using the Murnaghan-Nakayama formula. This is called as c:=CharacterTable("symmetric",5); (or replace 5 by your favorite degree)

Then ClassParameters(c) and CharacterParameters(c)are lists that give the associated partitions for the classes and the characters, e.g. the 4th entry is [1,[3,1,1]], indicating one character parameterized by the partition $1^23$.