[GAP Forum] decomposing regular representation + (AREP package)

[Dmitrii Pasechnik]

Dear R.N.Tsai, dear forum,

the main problem here is to obtain the complete list of irreducible complex
representations of G. IMHO computing the latter is a rather nontrivial
question for arbitrary finite groups.
Once you know such a list,
computing F is straightforward. See e.g. Sect. 2.7 of
J.-P. Serre, Linear representations of finite groups. Graduate
Texts in Mathematics, Vol. 42, Springer, New York, 1977

I have some GAP4 code (that I can make available upon request, although it's
not in any kind of polished form) that implements these formulae; we used it
in just completed preprint
http://www.ntu.edu.sg/home/dima/papers/truss6.pdf
where you can also find these formulae from Serre's book.
Actually, there it is used to compute a decomposition into irreducibles, and
then the centraliser ring, of a finite group representation, so it's a
slightly more general problem than yours.

Best,
Dmitrii
http://www.ntu.edu.sg/home/dima/

On 2/24/07 5:54 PM, "R.N. Tsai" <r_n_tsai at yahoo.com> wrote:

> Dear gap forum,
>    
>   I would like to decompose the regular (permutation) representation of
> some small groups into irreducible representations (over the complexes).
>    
>   That is for finite group G of order |G|, I would like an explicit
> |G|x|G| matrix F such that
>    
>     F^-1 R(g) F = B(g)
>    
>   R(g) is the regular representation, B(g) is block diagonal.
>   R(g),B(g) and F are all |G|x|G| matrices over complexs.
>    
>   Is there anything in GAP that would facilitate getting such a matrix
> explicitly?
>   
> I ran accross a GAP3 package "AREP" but I'm not sure if that has what I
> need (I didn't read through all its documentation yet); it also doesn't look
> like it's supported by GAP4 anyway, so it may not be easily usable even if
> it did.
>    
>   Thanks for your help.
>    
>   R.N.