Uniqueness of Branching and Unique Factorization of Tensor Products of Typical Representations of Lie Superalgebras

A theorem of Rajan states that a tensor product of irreducible, finite-dimensional representations of a simple Lie algebra over a field of characteristic zero uniquely determines its individual constituents. This is analogous to the uniqueness of prime factorization of natural numbers. We discuss a more general problem of determining all pairs ((V_1, V_2)) of finite-dimensional irreducible representations of a semisimple Lie algebra (\mathfrak g) such that $Res_{\mathfrak g_0} V_1 \cong Res_{\mathfrak g_0} V_2, where (\mathfrak g_0) is the fixed-point subalgebra of (\mathfrak g) with respect to a finite-order automorphism. We also discuss the analogous tensor product problem in the category of typical representations of basic classical Lie superalgebras.