Classifications of $\Gamma$-colored minuscule posets and $P$-minuscule Kac--Moody representations

The $\Gamma$-colored $d$-complete and $\Gamma$-colored minuscule posets unify and generalize multiple classes of colored posets introduced by R.A. Proctor, J.R. Stembridge, and R.M. Green. In previous work, we showed that $\Gamma$-colored minuscule posets are necessary and sufficient to build from colored posets certain representations of Kac--Moody algebras that generalize minuscule representations of semisimple Lie algebras. In this paper we classify $\Gamma$-colored minuscule posets, which also classifies the corresponding representations. We show that $\Gamma$-colored minuscule posets are precisely disjoint unions of colored minuscule posets of Proctor and connected full heaps of Green. Connected finite $\Gamma$-colored minuscule posets can be realized as certain posets of coroots in the corresponding finite Lie type.