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

[Dima Pasechnik]

Hi Vincent,
On Sat, Apr 11, 2020 at 08:05:35PM +0200, Vincent Delecroix wrote:
> 
> 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)?

Do you also have d given at this moment? If not, one can argue
that for d=|G| one has that (*2) holds automatically:
indeed, take the regular permutation representation of G. Then the
QQ[sqrt(5)]-irreducible will be there, and H={1_G} will certainly
leave every vector there invariant.

HTH.

Best
Dima

> 
> Best
> Vincent
> 
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