Constructing new polynomials by product of roots

Question

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.

Answer 1

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?.

Answer 2

They appear in the definition of (special) $\lambda$-rings, more precisely in the formula

$\lambda^k\left(\lambda^j\left(x\right)\right) = P_{k,j}\left(\lambda^1\left(x\right),\lambda^2\left(x\right),...,\lambda^{kj}\left(x\right)\right)$,

where $P_{k,j}\in \mathbb Z\left[\alpha_1,\alpha_2,...,\alpha_{kj}\right]$ is the polynomial which is, more or less, the $k$-th coefficient of what you call $P_j\left(t\right)$ written out as a polynomial in the coefficients of $P_1\left(t\right)$ (not in the $x_i$). Okay, not exactly your $P_j\left(t\right)$, but instead $\prod\limits_{j=1}^n\left(1+x_jt\right)$ (this is your $P_j\left(t\right)$ "written upside down").

There are several texts on $\lambda$-rings nowadays; here are two:

Donald Knutson, $\lambda$-Rings and the Representation Theory of the Symmetric Group, New York 1973.

Donald Yau, Lambda-rings, WS 2010. (Here you can find the first chapter which contains the definitions.)

You can probably pick any text (as long as it's not Fulton-Lang...), but keep in mind that what some texts call $\lambda$-ring is what others call special $\lambda$-ring. Also, your polynomials $P_k$ are not what is traditionally called $P_k$ in $\lambda$-ring theory.