Polynomial invariants of an S3 doublet through degree four ($\mathbb{C}[x,y{]}^{S_3}=\mathbb{C}[xy,x^3+y^3]$)

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 $\alpha \ne 1$. 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 $x^3{y}^3=(xy{)}^3$, by the recurrence $s_k=(x^3+y^3)s_{k-1}-(xy{)}^3{s}_{k-2}$. 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.