Simplification Rules for Birdtrack Operators

TL;DR

This paper introduces simplification rules for birdtrack operators involving symmetrizers and antisymmetrizers, enabling easier calculations and more compact representations in the context of SU(N) representation theory.

Contribution

It presents new cancellation and propagation rules for birdtrack operators, facilitating the construction of Hermitian Young projection operators.

Findings

Rules enable shortening of birdtrack expressions.

Propagation rules allow symmetrizer and antisymmetrizer interchange.

Application demonstrated in SU(N) representation calculations.

Abstract

This paper derives a set of easy-to-use tools designed to simplify calculations with birdtrack op- erators comprised of symmetrizers and antisymmetrizers. In particular, we present cancellation rules allowing one to shorten the birdtrack expressions of operators, and propagation rules identifying the circumstances under which it is possible to propagate symmetrizers past antisymmetrizers and vice versa. We exhibit the power of these simplification rules by means of a short example in which we apply the tools derived in this paper on a typical operator that can be encountered in the representation theory of SU(N) over the product space $V^{\otimes m}$ . These rules form the basis for the construction of compact Hermitian Young projection operators and their transition operators addressed in companion papers.