Given a partition$\lambda=(\lambda_1,\lambda_2,\cdots,\lambda_n)$, where $\lambda_1\geqslant\lambda_2\geqslant\cdots\lambda_n$, and each $\lambda_j$ is a non-negative integer, the Vandermonde matrix with deleted rows can be defined by
$\mathbf{V}=\left[ \begin{matrix} x_1^{\lambda_1+n-1} & x_2^{\lambda_1+n-1} & \dots & x_n^{\lambda_1+n-1} \\ x_1^{\lambda_2+n-2} & x_2^{\lambda_2+n-2} & \dots & x_n^{\lambda_2+n-2} \\ \vdots & \vdots & \ddots & \vdots \\ x_1^{\lambda_n} & x_2^{\lambda_n} & \dots & x_n^{\lambda_n} \end{matrix} \right]$ .
The determinant of the matrix $\mathbf{V}$ can be determined by the Schur polynomial. However, from the Jacobi−Trudi identities, the determinant of the matrix $\mathbf{V}$ is just expressed as a determinant in terms of the elementary symmetric polynomials, which is still hard to be checked whether the determinant is zero or not. Is there any way that can check whether the matrix $\mathbf{V}$ is full rank or not?
Specifically, I assume that $x_i=e^{j2\pi\frac{i}{n}}$, $\forall i\in [1,2,\cdots,n]$.