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As an exercise for myself, I've been trying for some time to calculate the coordinate ring of the $E6$ surface singularity. That is, I have a 2d representation of the binary tetrahedral group generated by

$(x,y) \rightarrow (i x, i y)$

$(x,y) \rightarrow (-y, x)$

$(x,y)\rightarrow (\frac{\xi}{\sqrt{2}}(x+y),\frac{\xi}{\sqrt{2}}(x-y))$,

where $\xi$ is a primitive 8th root of unity. I want to calculate the subring of invariants of this group action in $\mathbb{C}[x,y]$.

Some things I already know include the answer (no proof of this): $\mathbb{C}[X,Y,Z]/(X^2+Y^3+Z^4)$

and a couple invariants : $(x^2+y^2)^4$ and $(x+y)^8 + (x-y)^8 -16x^8 - 16y^8$ (hopefully I have the signs right here).

I calculated both of these invariants by messing around with the transformation in sage. Maybe there is a way that sage knows how to calculate these invariants?

Anybody have any ideas?

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There are algorithms, based on Groebner bases, which compute invariants and their syzygies; you van do this with Macaulay, IIRC. – Mariano Suárez-Alvarez Dec 6 '12 at 3:19
    
A sensible approach is to compute the Hilbert series of the invariant subring (using Molien's formula) and some linear algebra to find new invariants. – Mariano Suárez-Alvarez Dec 6 '12 at 3:20

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