Consider the general polynomial $P_1(t) = \prod_{j=1}^n (t+x_j)$. Construct $P_k(t) = \prod_{\sigma \subset [n], |\sigma|=k} (t+x_{\sigma_1}x_{\sigma_2}\cdots x_{\sigma_k})$ where the product is over all subsets of size $k$ of the numbers $1,2,\dots,n.$ The coefficients of $P_1(t)$ will be the elementary symmetric polynomials in $x_1,\dots,x_n$ and it is easy to argue that the coefficients in $P_k(t)$ are polynomials in the coefficients of $P_1.$

Now, my question is rather vague but I seek references to areas where these types of polynomials appear. I suspect they are related somehow to Schur-polynomials, representation theory, and determinants of band-matrices.