  The full exploitation of non-abelian symmetries in tensor network states
(TNS) derived from a given lattice Hamiltonian is highly attractive in various
aspects. From a theoretical perspective, it can offer deep insights into the
entanglement structure and quantum information content of strongly correlated
quantum many-body states. From a practical perspective, it allows one to push
numerical efficiency by orders of magnitude. Physical expectation values based
on TNS require the full contraction of a given tensor network, with the
elementary ingredient being a pairwise contraction. While well-established for
no or just abelian symmetries, this can become quickly extremely involved and
cumbersome for general non-abelian symmetries. As shown in this work, however,
the latter can be tackled in a transparent and efficient manner by introducing
so-called X-symbols which deal with the underlying pairwise contraction of
generalized Clebsch-Gordan tensors (CGTs). These X-symbols can be computed
deterministically once and for all, and hence also be tabulated. Akin to
6j-symbols, X-symbols are generally much smaller than their constituting CGTs.
In applications, they solely affect the tensors of reduced matrix elements, and
therefore, once tabulated, allow one to completely sidestep the explicit usage
of CGTs, and thus to greatly increase numerical efficiency.
