  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.
