rather basic question here that's coming up out of curiosity, motivated by understanding the restriction and induction functors in full detail for the alternating subgroup of the symmetric group.
My question is the following: if two representations of $S_n$ restrict to the same representation of $A_n$, does it follow that each equals the tensor of the other one by the alternating representation? My first thought is that, a fortiori, does anybody know if two tableau having the same product of hook lengths, implies that the tableaux are conjugate? (Of course then hook length formula implies my hypothesis can be weakened to saying any two irreducibles of the same dimension of $S_n$).
Edit: I've found that my stronger guess about dimensions is resoundingly false: https://mathoverflow.net/questions/123690/dimension-of-specht-modules-s-lambda