# [GAP Forum] decomposition formulae for C-representations

David Joyner wdj at USNA.Navy.Mil
Wed Apr 7 15:54:35 BST 2004

```Dmitrii Pasechnik wrote:

>Dear Forum,
>
>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.
>For instance to obtain the subrepresentation of p (a direct sum of
>irreducible representations with the same character chi) of dimension
>n_chi, corresponding to the irrducible character chi, one uses the projection
>	n_chi/|G| sum_{g in G} chi^*(g)p(g)
>(Thm. 8 in Sect. 2.6 of the Serre's book)
>
>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.
>
>thanks
>Dmitrii
>
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>

How is your repn p entered? For example,

A5:=AlternatingGroup(5);
A5_repns:=Irr(A5);
C:=Centralizer(A5,(1,2)(3,4));
C_repns:=Irr(C);
ind:=InducedClassFunction(C_repns[1],A5);
m:=List(A5_repns,x->ScalarProduct(ind,x));

returns the multiplicities occuring in an
induced repn.

If p is entered as a class function, I'm not sure,
but I'd like to know too!

```