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Multivariate polynomial indexed by ${1, \ldots, n}$ are acted on by $S_n$: for $\sigma \in S_n$, define $\sigma(x_i) = x_{\sigma(x_i)}$, etc. Symmetric polynomials are those polynomials which are invariant under this action.

If we consider just a subgroup of $S_n$, a polynomial which is not symmetric my be invariant. An example: Let $S_5$ be the symmetric group of degree 5, $\tau = (1\ 2\ 3\ 4\ 5)$, and $C_5 = \{\tau^n : n = 1, ..., 5\}$. The polynomial $$f(x_1, x_2, x_3, x_4, x_5) = x_1x_3 + x_2x_4 + x_3x_5 + x_4x_1+x_5x_2$$ is invariant under the elements of $C_5$, but not under the elements of $S_5$.

I was wondering what sort of theory exists around these sets of polynomials? What are they called?

Also, Every symmetric polynomial can be expressed as some polynomial evaluated on the elementary symmetric polynomials. Is there some systematic way of generating some set of polynomials which will have this property?

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    $\begingroup$ "Invariant theory" is one keyword (well, key-phrase) for getting into the large literature on this kind of question. $\endgroup$ – Noam D. Elkies Aug 15 at 2:48
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    $\begingroup$ It's probably worth pointing out that the $S_n$ may act on polynomial rings in other interesting ways. For doubly indexed variables the action $\sigma \cdot x_{i\, j} = x_{\sigma (i) \, \sigma (j)}$ plays a role in the study of graph density inequalities. $\endgroup$ – tim Aug 15 at 3:11
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    $\begingroup$ See the following MO question from 2010: mathoverflow.net/questions/14613/…. $\endgroup$ – KConrad Aug 15 at 3:47
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    $\begingroup$ For some work in this area see sciencedirect.com/science/article/pii/0001870884900057. $\endgroup$ – Richard Stanley Aug 15 at 14:39
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    $\begingroup$ See also Richard Stanley's paper Invariants of finite groups and their applications to combinatorics, projecteuclid.org/euclid.bams/1183544328. $\endgroup$ – Ira Gessel Aug 18 at 5:28
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If you're interested specifically in the cyclic group you may be interested in the "cyclic quasi-symmetric functions" studied recently in this paper of Adin, Gessel, Reiner, and Roichman: https://arxiv.org/abs/1811.05440. These "cyclic quasi-symmetric functions" are not exactly the same as polynomials invariant under the action of the long cycle in $S_n$ (for instance, they are formal power series rather than polynomials, and also they also have to be "quasi-symmetric functions" in addition to enjoying a cyclic symmetry condition). But the theory here closely parallels the usual theory of symmetric functions (and the theory of quasi-symmetric functions) and does involve a kind of cyclic symmetry.

By the way, the cyclically symmetric polynomials (in your sense) were also discussed in this question of Jim Propp: Cyclically symmetric functions.

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