= Polynomial invariants of an S3 doublet through degree four
{title2=$\mathbb C[x,y]^{S_3}=\mathbb C[xy,x^3+y^3]$}
Let the <symmetric group> $S_3$ act by $(x,y)\mapsto(\alpha x,\alpha^{-1}y)$ and $(x,y)\mapsto(y,x)$, where $\alpha^3=1$ and $\alpha\ne1$. Rotation invariance requires the exponents of each <monomial> to differ by a multiple of three. After symmetrizing, factor out a power of $xy$; the remainder is $x^{3k}+y^{3k}$. These power sums are <polynomials> in $x^3+y^3$ and $x^3y^3=(xy)^3$, by the recurrence $s_k=(x^3+y^3)s_{k-1}-(xy)^3s_{k-2}$. Hence the entire <polynomial invariant ring> is generated by $xy$ and $x^3+y^3$, which are algebraically independent. Their degrees are two and three, giving one invariant in each of degrees two, three and four, and none in degree one.
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