Representations by Young Diagrams. The Method of Irreducible Tensors

Abstract

In Sect. 5.5.3 we described how to reduce tensor spaces. From this we are able to obtain the basis functions belonging to the IRs D ^[λ](n) of GL(n, ℂ) and of its subgroups U(n), LU(n), O(n), etc. The basis functions which span the IR spaces $\mathcal{L}_{{n,k}}^{{\left[ \lambda \right]}}$ of D ^[λ](n) are according to (5.5.31) the symmetrized irreducible tensors of rank m $\Psi_{{f,k}}^{{\left[ \lambda \right]}}$ with f = {f ₁,…, f _m }. This description of the IRs of continuous groups is quite different from the Lie-Cartan-Weyl method; it supplements the latter. In order to illustrate this, we recall the method for finding the basis tensors.