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?

Invariants of finite groups and their applications to combinatorics, projecteuclid.org/euclid.bams/1183544328. $\endgroup$ – Ira Gessel Aug 18 at 5:28