Let $\lambda$ be a partition of $n$, and $\lambda^*$ the dual partition (i.e. having the transposed Young diagram). Let $z_i$ be vectors in $\mathbb{C}^{\lambda_i^*}$, and$$F_\lambda = \prod_i \Delta_{\lambda_i^*}(z_i),$$where $\Delta_m(x) = \prod_{1 \le i < j \le m} (x_i - x_j)$ is the Vandermonde determinant. Then $F_\lambda$ is a polynomial in $N$ variables (say, $x_1, \dots, x_n)$. Let $W_\lambda$ is the linear span of the $S_n$-translates of $F_\lambda$. Show that $W_\lambda$ is isomorphic to the Specht module $V_\lambda$ of $S_n$.
Does anyone have a reference to this seemingly standard fact of representation theory? Thanks.