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.



Constructing new polynomials by product of roots
In the language of symmetric functions, you are computing the *plethysm* $e\_j[e\_k]$ (also denoted $e\_k\circ e\_j$) of elementary symmetric functions. In terms of $\mathrm{GL}(n,\mathbb{C})$ representations, you are looking at the representation $\Lambda^j(\Lambda^k(\mathbb{C}^n))$. Some information is at [What is known about this plethysm?](https://mathoverflow.net/questions/23391). 


