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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