[GAP Forum] transitive groups with irreducible representations defined over QQ[sqrt(5)]

[Thomas Breuer]

Dear GAP Forum,

property (*2) for the irreducible representation \rho of G means
that the restriction of \rho to H has a trivial constituent 1_H
and thus \rho is a constituent of (1_H)^G, which is the natural
permutation representation of G.

(This solves the problem if G is 2-transitive.)

The following answer does not fit to the three questions,
but it may be interesting.
There is an algorithm by Michler and Weller for computing the
abovementioned constituents, see

    G. O. Michler and M. Weller, 
    The character values of the irreducible constituents
    of a transitive permutation representation, 
    Arch. Math. (Basel) 78 (2002), no. 6, 417–429. 

I am not aware of a GAP implementation of this algorithm.

All the best,
Thomas


On Sat, Apr 11, 2020 at 08:05:35PM +0200, Vincent Delecroix wrote:
> Dear all,
> 
> I am looking for transitive groups G of S_d (I don't know d) with
> the following properties
> 
>  (*1) G admits an irreducible representation defined over
>       QQ[sqrt(5)]  (in particular the character is real)
> 
>  (*2) the stabilizer H of 1 in G admits invariant vectors
>      in that irreducible representation.
> 
> Question number one: knowing an irreducible representation defined
> over QQ[sqrt(5)] what is the fastest way to check for (*2)?
> 
> Question number 2: so far, I am computing the full character table
> for the group G and this is very expensive. Do you know of any
> practicable criterion that would allow me to discard groups without
> property (*1)?
> 
> Question number 3: given G is there a way to access the characters
> defined over QQ[E(5)] but not the one with higher conductors without
> filtering the list of irreducible characters?
> 
> Best
> Vincent
> 
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