Transition operators

In this paper, we give a generic algorithm of the transition operators between Hermitian Young projection operators corresponding to equivalent irreducible representations of $𝖲𝖴(N)$⁠, using the compact expressions of Hermitian Young projection operators derived in the work of Alcock-Zeilinger and Weigert [eprint arXiv:1610.10088 [math-ph]]. We show that the Hermitian Young projection operators together with their transition operators constitute a fully orthogonal basis for the algebra of invariants of $V⊗m$ that exhibits a systematically simplified multiplication table. We discuss the full algebra of invariants over $V⊗3$ and $V⊗4$ as explicit examples. In our presentation, we make use of various standard concepts, such as Young projection operators, Clebsch-Gordan operators, and invariants (in birdtrack notation). We tie these perspectives together and use them to shed light on each other.