Z_1$. In [@donaldson5] Donaldson computes these values theoretically, and our investigations corroborate his result: $$\label{eqn: sigma for CP1}
    \sigma_{T_{\nu},k} =
        \left\{ \begin{array}{rl}
            \frac{(k-1)k}{(k+2)(k+3)} & \mathrm{if\ }G\mathrm{\ is\ inv.\ under\ }Z_0 \leftrightarrow Z_1 \\
            \frac{k}{(k+2)} & \mathrm{otherwise}
        \end{array} \right. .$$ Our goal is to show evidence for a simple extension of this formula valid on $X={\mathbb{CP}}^n,\ n\geq 1$.

When $n>1$ there are many possible ways to extend the notion of a “palindromic” metric (as we defined in section \[section: The T operators\]): for the Riemann sphere we have those metrics invariant under $Z_0 \leftrightarrow Z_1$ but in general there are many permutations of the homogeneous coordinates $Z_0, \ldots, Z_n$ and it is trivial to check that if $G$ is a metric invariant under such a symmetry then so is ${T_{\nu}}(G)$. We might then expect that there can be distinct values for $\sigma$ depending on various symmetries the metric $G$ could be invariant under. Thus we may find a different value for each (conjugacy class of) subgroup of $\mathrm{Sym}(n+1)$ – the symmetric group on $n+1$ characters – corresponding to metrics $G$ invariant under the automorphisms $Z_i \mapsto Z_{\pi(i)},\ i=0, 1, \ldots, n$ for $\pi$ ranging in the subgroup.

We present here some 