[GAP Forum] Re: decomposition formulae for C-representations

[Thomas Breuer]

Dear GAP Forum,

Dima Pasechnik asked

> are the formulae that give the decomposition of a C-represenation p
> of a finite group G into direct sums of irreducibles implemented in GAP?
> (perhaps somewhere within character theory machinery I guess)
> 
> I mean the standard ones given in e.g. Serre's "Linear Representations
> of Finite Groups", Sect. 2.6 and 2.7.
> [...]
> Certainly, this is only feasible for groups of relatively small order
> to use these formulae directly, but in our case the groups
> are of order <10^4.

I am not aware of a GAP implementation for that.
One possibility for groups of this small order would certainly be
to compute the character of the given representation,
to decompose it into irreducibles,
to compute all irreducible representations of the group,
and to use their characters for deciding which ones occur as summands
of the given representation.

Of course this can be viewed as an ugly attempt because it does not
really decompose the given data.
But computing the irreducible representations of small groups is not hard,
and for the case of abelian by supersolvable groups one gets them in the
nice form of monomial representations.

The question is whether this suffices or whether you need the explicit
base change that transforms the given matrices into block diagonal form.
(And of course another question is whether it would be possible to avoid
computations with representations at all.)

All the best,
Thomas