Transition Operators

TL;DR

This paper introduces a generic algorithm for transition operators between Hermitian Young projection operators for equivalent irreducible representations of SU(N), simplifying the algebra of invariants in tensor spaces.

Contribution

It provides a systematic method to construct transition operators and demonstrates their role in forming an orthogonal basis for the algebra of invariants, with explicit examples for tensor powers.

Findings

Transition operators form an orthogonal basis for the algebra of invariants.

Simplified multiplication table for the algebra of invariants.

Explicit examples for tensor powers V^{⊗3} and V^{⊗4}.

Abstract

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 a companion paper. We show that the Hermitian Young projection operators together with their transition operators constitute a fully orthogonal basis for the algebra of invariants of $V^{\otimes m}$ that exhibits a systematically simplified multiplication table. We discuss the full algebra of invariants over $V^{\otimes 3}$ and $V^{\otimes 4}$ 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.