Relative scale separation in orbifolds of S2 and S5

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