I'm now reading 1204.2230 and 1512.07213 on K-stability as this is sort of related to superconformal field theory in physics. In these papers, the Hilbert series is used to compute the Futaki invariant. However, I'm not sure how to find the allowed test configurations/symmetries.
So for example, if we consider the ring $\mathbb{C}[u,v,y,z]/I$ with $I$ generated by $uv+y^2+z^m=0$. We can consider the test symmetry with weights $(0,0,0,1)$ and compute its Futaki invariant from the Hilbert series, which shows that the Futaki invariant is only positive for $m=2,3$ ($m=4$ gives zero Futaki invariant, but has a non-zero norm).
However, if we consider the test symmetry with weights $(1,-1,0,0)$ (though we wouldn't have this if we make a linear change to $w^2+x^2+y^2+z^2=0$), we would get zero Futaki invariants and non-zero norms for all $m$, but this test configuration should be trivial. Moreover, we can consider the one with weights like $(-1,-1,-1,\frac{6}{m})$ (for $w^2+x^2+y^2+z^m=0$). This would lead to being K-unstable for all $m$. However, as a subcase of BP singularity, it is shown in 1512.07213 that it should be K-stable when $m\leq3$, i.e., $m=2,3$.
So I'm not sure what kind of test configurations/symmetries are allowed. I understand that from the above two papers, we require that the test configuration should be normal. They also say that we should find the new $\mathbb{C}^*$-action that commutes with the original torus action $T$. However, I don't think I really understand in what sense the commutation means. Naively, as we always assume diagonal actions (as the above examples), we would consider diagonal matrices (which commute with each other). But it does not seem to be right as the above examples are diagonal actions/matrices (satisfying normality).
Would anyone let me know in what sense this commutation means? It would be very helpful if you could give any example/counterexample.
Also, I'm wondering if there's any further constraint on the allowed test configuration (besides normality and commutation). Do we need anything else such as being orthogonal to $T$, i.e., K-stability relative to $T$? (I tried this but it seems to be too strict than the results in the above papers.) Thank you so much!