General indices of simple Lie algebras and symmetrized product representations

Abstract

In many branches of physics, it is important to know the decomposition of a product representation ρ⊗ρ⊗⋅⋅⋅⊗ ρ (n times) of identical representations ρ of a simple Lie algebra into irreducible components with a given Young tableau symmetry. We show that the notion of representation indices introduced elsewhere is a very useful tool in dealing with this problem. We calculate explicit formula for general pth order indices D( p) ( ρ) for all classical simple Lie algebras. Sixth-order indices for exceptional Lie algebras are also discussed.