Is there any thing in the literature that discusses zeros of Schur functions over $\mathbb{C^n}$? When i say Schur function I mean the one you obtain by dividing the generalized Vandermonde determinant by the principle Vandermonde determinant. Here the generalized Vandermonde determinant is one which has $m_1<\dots < m_n$ being a set of increasing integers as exponents of the entries, and all $m_i>0$. This Generalized Vandermonde is found in this paper http://matwbn.icm.edu.pl/ksiazki/aa/aa95/aa9522.pdf

Another question would be is there a way of imposing some group theortic structure on the zeros of schur functions over $\mathbb{C^n}$ as a consequence of the symmetry of the zeros (because if $(z_1,\dots , z_n)$ is a zero then all permutations thereof are zeros as well by symmetry of the schur function over all of its variables).