Is there an account of the algebra of highest weight tensors? 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.
 A: The highest weight vectors in the tensor algebra are obtained by applying certain Young symmetrizers to tensor products of standard basis vectors of $V$. For example, a highest weight vector is obtained from
$$
e_1 \otimes e_1 \otimes e_2 \otimes e_2
$$
by applying the Young symmetrizer of the tableau of shape $(2,2)$ filled with $1,2,3,4$ from left-to-right top-to-bottom. There is a basis for the space of highest weight vectors indexed by pairs of tableaux $(T,S)$ where $T$ and $S$ have the same shape, row $i$ of $T$ consists of the number $i$ and $S$ is standard. I think this is written up in Fulton's book on Young Tableaux.
The product of two highest weight vectors must be "straightened" to be expressed as a integer linear combination of highest weight vectors of this form. This straightening algorithm is well know (in complete generality by Grosshans--Rota--Stein, but this case might be due to De Concini and Procesi, perhaps {way} eariler. You might look in Brian Taylor's MIT thesis to see that such relations form a Groebner basis).
