[GAP Forum] symmetric powers of a character

[Thomas Breuer]

Dear GAP Forum,

Dima Pasechnik asked

> is there a GAP function (or available GAP code) to compute
> symmetric powers of a complex character?
> (As far as I understand, such code would require manipulating
> symmetric functions...)

If I understand the question correctly then we are given
a positive integer $k$ and a complex character $\chi$
that is afforded by the module $V$, say,
and we want to compute the character of the fixed subspace of the
$k$-fold tensor power of $V$ w.r.t. the natural action
of the symmetric group on $k$ letters (permuting the components
of the tensors).

The library function `SymmetricParts' can be used for that,
see "Symmetrizations of Class Functions"
    http://www.gap-system.org/Manuals/doc/htm/ref/CHAP070.htm#SECT011
in the GAP Reference Manual.

Here is an example.

    gap> t:= CharacterTable( "A5" );
    CharacterTable( "A5" )
    gap> chi:= Irr( t )[4];
    Character( CharacterTable( "A5" ), [ 4, 0, 1, -1, -1 ] )
    gap> List( [ 1 .. 10 ], k -> SymmetricParts( t, [ chi ], k )[1] );
    [ Character( CharacterTable( "A5" ), [ 4, 0, 1, -1, -1 ] ),
      Character( CharacterTable( "A5" ), [ 10, 2, 1, 0, 0 ] ),
      Character( CharacterTable( "A5" ), [ 20, 0, 2, 0, 0 ] ),
      Character( CharacterTable( "A5" ), [ 35, 3, 2, 0, 0 ] ),
      Character( CharacterTable( "A5" ), [ 56, 0, 2, 1, 1 ] ),
      Character( CharacterTable( "A5" ), [ 84, 4, 3, -1, -1 ] ),
      Character( CharacterTable( "A5" ), [ 120, 0, 3, 0, 0 ] ),
      Character( CharacterTable( "A5" ), [ 165, 5, 3, 0, 0 ] ),
      Character( CharacterTable( "A5" ), [ 220, 0, 4, 0, 0 ] ),
      Character( CharacterTable( "A5" ), [ 286, 6, 4, 1, 1 ] ) ]

For the next version of GAP, I will add a few remarks and index entries,
such that it is easier to find what Dima had asked for.

All the best,
Thomas