Many methods exist for the construction of the Hilbert series describing the
moduli spaces of instantons. We explore some of the underlying group theoretic
relationships between these various constructions, including those based on the
Coulomb branches and Higgs branches of SUSY quiver gauge theories, as well as
those based on generating functions derivable from the Weyl Character Formula.
We show how the character description of the reduced single instanton moduli
space of any Classical or Exceptional group can be deconstructed faithfully in
terms of characters or modified Hall-Littlewood polynomials of its regular
semi-simple subgroups. We derive and utilise Highest Weight Generating
functions, both for the characters of Classical or Exceptional groups and for
the Hall-Littlewood polynomials of unitary groups. We illustrate how the root
space data encoded in extended Dynkin diagrams corresponds to relationships
between the Coulomb branches of quiver gauge theories for instanton moduli
spaces and those for T(SU(N)) moduli spaces.Comment: 97 pages, 12 figure