  A group-theoretical structure in a perturbative expansion of the Wilson loops
in the 3d Chern-Simons theory with $SU(N)$ gauge group is studied in symmetric
approach. A special basis in the center of the universal enveloping algebra
$ZU(\mathfrak{sl}_N)$ is introduced. This basis allows one to present group
factors in an arbitrary irreducible finite-dimensional representation.
Developed methods have wide applications, the most straightforward and evident
ones are mentioned. Namely, Vassiliev invariants of higher orders are computed,
a conjecture about existence of new symmetries of the colored HOMFLY
polynomials is stated, and the recently discovered tug-the-hook symmetry of the
colored HOMFLY polynomial is proved.
