  We consider the problem of determining the relationship between two
representations knowing that some tensor or symmetric power of the original
represetations coincide. Combined with refinements of strong multiplicity one,
we show that if the characters of some tensor or symmetric powers of two
absolutely irreducible $l$-adic representation with the algebraic envelope of
the image being connected, agree at the Frobenius elements corresponding to a
set of places of positive upper density, then the representations are twists of
each other by a finite order character.
