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

TL;DR

This paper classifies $ ext{Gamma}$-colored minuscule posets, unifying various classes and linking them to representations of Kac--Moody algebras, thereby advancing the understanding of their structure and classification.

Contribution

It provides a complete classification of $ ext{Gamma}$-colored minuscule posets, connecting them to known poset classes and Lie algebra representations.

Findings

$ ext{Gamma}$-colored minuscule posets are disjoint unions of known classes.

Finite $ ext{Gamma}$-colored minuscule posets correspond to posets of coroots.

Classification of these posets determines the associated Kac--Moody representations.

Abstract

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.