Given a complex vector space V, consider the ring of polynomial
functions on the space of configurations of a vectors and b
covectors which are invariant under the natural action of SL(V).
Rings of this type play a central role in representation theory,
and their study dates back to Hilbert. Over the last three
decades, different bases of these spaces were found. To explicitly
construct, as well as to compare, some of these remarkable bases
remains a challenging problem, already open when V is 3-dimensional.
In this talk, I report on recent developments in this theory in
the 3-dimensional setting.