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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.

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  • $\begingroup$ I don't see how $G_N^M$ are symmetric, unless I misunderstood the hypotheses. Minor point: you seem to be using very non-standard notation for what is a very standard object in symmetric functions theory. $\endgroup$
    – user35313
    Jul 29, 2020 at 17:01
  • $\begingroup$ Edited. Thanks! $\endgroup$
    – twofiveone
    Jul 29, 2020 at 17:20
  • $\begingroup$ The polynomial is still not symmetric, unless all the $S $es are equal. $\endgroup$ Jul 29, 2020 at 17:27
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    $\begingroup$ As @darijgrinberg implies, I think you may be running into the confusion about "how symmetric". A symmetric polynomial is symmetric in all its variables, whereas a symmetric tensor is symmetric only in the indices that are passed to it. (I'm also not sure what you gain by thinking of tensors here.) For example, any degree-1 tensor, say $S_i = i$ in the index notation, is symmetric; but $G^2_1 = X_1 + 2X_2$ is not a symmetric polynomial. $\endgroup$
    – LSpice
    Jul 29, 2020 at 17:41

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