On vector invariants of the symmetric group

The purpose of this paper is to give a proof of the results announced by the author [7] in 1982 on the algebraic independence over a field $k$ of any non-degenerate system of $mn$ distinct basis invariants in the ring $k[x_{11}, \dots ,x_{1n}; \dots ; x_{m1}, \dots , x_{mn}]$ with respect to the symmetric group $G=S_{n}$. The result of this paper can be extended to the case of an arbitrary finite group.
The work was partially supported by the Russian Foundation for Basic Research, Grant 94–01–01206–a.