Generalization of elementary symmetric polynomials

Question

The elementary symmetric polynomials (ESPs) are defined as -

\begin{align*} E_{1}^{1} &= X_1, \\ E_{1}^{2} &= X_1 + X_2, \\ E_{2}^{2} &= X_1 X_2, \\ E_{2}^{3} &= X_1 X_2 + X_1 X_3 + X_2 X_3, \\ E_{k}^{n} &= \sum_{1 \leq j_1 < \dotsb < j_k \leq n} X_{j_1} \dotsm X_{j_k}. \end{align*}

Notice that in the above examples, all the coefficients are 1.

Now we can generalize $E_{k}^{n}$ by the following function

$$ G_{k}^{n} = \sum_{1 \leq j_1 < \dotsb < j_k \leq n} S_{j_1 \cdots j_k} \: X_{j_1} \dotsm X_{j_k}, $$

where $S_{j_1 \cdots j_k}$ is symmetric. If $S_{j_1 \cdots j_k} = 1$, the above equation reduces to the ESP.

Edit: although $S_{j_1 \cdots j_k}$ is symmetric, $G_{k}^{n}$ is not (in the previous versions I incorrectly wrote $G_{k}^{n}$ is a symmetric polynomial).

My questions are -

1. Does the polynomial $G_{k}^{n}$ have a name?

2. Do decompositions of $G_{k}^{n}$ exist, like the ESP (by Lee - Power sum decompositions of elementary symmetric polynomials, 2016)?

I apologize if my notations are not standard. I have edited some of them to look familiar.