  Certain star shaped quivers exhibit a pattern of symmetry enhancement on the
Coulomb branch of $3d$ $\mathcal{N}=4$ supersymmetric gauge theories. This
paper studies a subclass of theories where such global symmetry enhancement
occurs through a computation of the Highest Weight Generating Function (HWG)
and of the corresponding Hilbert Series (HS), providing a further test of the
Coulomb branch formula. This special subclass has a feature in which the HWG
takes a particularly simple form, as a simple rational function which is either
a product of simple poles (termed freely generated) or a simple PE (termed
complete intersection). Out of all possible star shaped quivers, this is a
particularly simple subclass. The present study motivates a further study of
identifying all star shaped quivers for which their HWG is of this simple form.
