  Let $G={\rm GL}_n$ be the general linear group over an algebraically closed
field $k$, let $\mathfrak g=\mathfrak gl_n$ be its Lie algebra and let $U$ be
the subgroup of $G$ which consists of the upper uni-triangular matrices. Let
$k[\mathfrak g]$ be the algebra of polynomial functions on $\mathfrak g$ and
let $k[\mathfrak g]^G$ be the algebra of invariants under the conjugation
action of $G$. In characteristic zero, we give for all dominant weights
$\chi\in\mathbb Z^n$ finite homogeneous spanning sets for the $k[\mathfrak
g]^G$-modules $k[\mathfrak g]_\chi^U$ of highest weight vectors. This result
(with some mistakes) was already given without proof by J.~F.~Donin. Then we do
the same for tuples of $n\times n$-matrices under the diagonal conjugation
action. Furthermore we extend our earlier results in positive characteristic
and give a general result which reduces the problem to giving spanning sets of
the highest weight vectors for the action of ${\rm GL}_r\times{\rm GL}_s$ on
tuples of $r\times s$ matrices. This requires the technique called
"transmutation" by R.~Brylinsky which is based on an instance of Howe duality.
In the cases that $\chi_{{}_n}\ge -1$ or $\chi_{{}_1}\le 1$ this leads to new
spanning sets for the modules $k[\mathfrak g]_\chi^U$.
