Simplification rules for birdtrack operators

This paper derives a set of easy-to-use tools designed to simplify calculations with birdtrack operators 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 $V⊗m$⁠. These rules form the basis for the construction of compact Hermitian Young projection operators and their transition operators addressed in companion papers [J. Alcock-Zeilinger and H. Weigert, “Compact Hermitian Young projection operators,” e-print arXiv:1610.10088 [math-ph] and J. Alcock-Zeilinger and H. Weigert, “Transition operators,” e-print arXiv:1610.08802 [math-ph]].