The context for this question is Schur-Weyl duality. Let $V$ be a vector space. For $r>0$ consider $\otimes^rV$. This has commuting actions of the symmetric group $S(r)$ and $G=GL(V)$.

My question is prompted by reading $\S 51$ in "Compact Lie groups and their representations" by D. P. Zelobenko.

Choose a Borel subgroup $B\subset G$ and let $N\subset B$ be the unipotent radical. Then by the algebra of highest weight tensors I mean the algebra of $N$-invariant tensors in the tensor algebra of $V$. Is there a description of this algebra? say, generators, relations and a basis indexed by standard tableaux?

Since I expect this to be known (perhaps implicitly) a supplementary question is whether the $q$-analogue of this algebra is a deformation of the plactic monoid algebra? More precisely it should be possible to put $q=0$ to get the plactic monoid algebra.