Reduction of tensor products with definite permutation symmetry: Embeddings of irreducible representations of Lie groups into fundamental representations of SU(M) and branchings

Abstract

We consider tensor products made out of a number of identical copies of the defining representations of Lie groups that are asymptotically free and complex. Decomposition of the tensor products into the terms with definite permutation symmetry is made by using the index sum rules and the congruence class. The results can also be used to find the branchings of SU(M) into a Lie group G, where M is equal to the dimension of the defining representation of G. Application of our results to preon dynamics is indicated in two examples.