Relative scale separation in orbifolds of $S^2$ and $S^5$

In orbifold vacua containing an $S^q/\Gamma$ factor, we compute the relative order of scale separation, $r$, defined as the ratio of the eigenvalue of the lowest-lying $\Gamma$-invariant state of the scalar Laplacian on $S^q$, to the eigenvalue of the lowest-lying state. For $q=2$ and $\Gamma$ finite subgroup of $SO(3)$, or $q=5$ and $\Gamma$ finite subgroup of $SU(3)$, the maximal relative order of scale separation that can be achieved is $r=21$ or $r=12$, respectively. For smooth $S^5$ orbifolds, the maximal relative scale separation is $r=4.2$. Methods from invariant theory are very efficient in constructing $\Gamma$-invariant spherical harmonics, and can be readily generalized to other orbifolds.