[GAP Forum] Recombining irreducible representations

[Thomas Breuer]

Dear GAP Forum,

Jerry Swan asked

> For some element g of a group G for which irr :=
> IrreducibleRepresentations(G) have been obtained, is it possible to
> recover
> g from images := List(irr,r->Image(r,g)) ?

and later added

> Concretely, I'm working with $S_n$, which I should have clarified.

If phi is a group homomorphism and you know phi( g ) for some
group element g in the source of phi
then the set of preimages of phi( g ) under phi consists of the
coset g ker( phi ).

If phi is faithful (and S_n has faithful irreducible representations)
then just take the unique preimage under phi.

In general, if you know the images of g under sufficiently many
(irreducible) representations phi_i such that the intersection
of their kernels is trivial then you can determine g
from the intersection of the preimages of the phi_i( g ).

All the best,
Thomas