Ranks of group-invariant sphere maps with source dimension greater than 2

We study the target ranks for monomial sphere maps from Cn to CN that are invariant under some finite subgroup Γ of the unitary group on Cn. Previous work has established, for each such Γ, the existence of a threshold beyond which all target ranks are possible. The sharp threshold has been identified for most groups in source dimension 2 but not for higher source dimension n. In this paper, we obtain significant improvements in the estimate of this threshold for n>2 that illustrate that the situations for n = 2 and n>2 are qualitatively similar.