Uniqueness of branching and unique factorization of tensor products of typical representations of Lie superalgebras

A theorem of Rajan says that a tensor product of irreducible, finite dimensional representations of a simple Lie algebra over a field of characteristic zero determines individual constituents uniquely. This is analogous to the uniqueness of prime factorization of natural numbers. We discuss a more general question of determining all the pairs (V1, V2) consisting of two finite dimensional irreducible representations of a semisimple Lie algebra g such that Res_g0 V1 = Res_g0 V2, where g0 is the fixed point subalgebra of g with respect to a finite order automorphism. We will also discuss the above tensor product problem in the category of typical representations of basic classical Lie superalgebras.