Unique decomposition of tensor products of irreducible representations of simple algebraic groups ByC. S. Rajan Abstract We show that a tensor product of irreducible, finite dimensional representations of a simple Lie algebra over a field of characteristic zero determines the individual constituents uniquely. This is analogous to the uniqueness of prime factorisation of natural numbers. 1. Introduction. 2. Preliminaries 3.GL(2) 4. Tensor products ofGL(r)-modules 5. Proof of the main theorem in the general case 6. An arithmetical application