Classifications of $\Gamma$-colored $d$-complete posets and upper $P$-minuscule Borel representations

TL;DR

This paper classifies $ ext{Gamma}$-colored $d$-complete posets, revealing their structure as minuscule heaps and order filters, which correspond to certain Borel representations in Lie theory.

Contribution

It provides a complete classification of finite and infinite $ ext{Gamma}$-colored $d$-complete posets, linking them to minuscule heaps and full heaps, extending previous combinatorial frameworks.

Findings

Finite $ ext{Gamma}$-colored $d$-complete posets are exactly the dominant minuscule heaps.

Infinite connected $ ext{Gamma}$-colored $d$-complete posets are order filters of connected full heaps.

The classification extends and unifies previous combinatorial models of minuscule representations.

Abstract

The $\Gamma$-colored $d$-complete posets correspond to certain Borel representations that are analogous to minuscule representations of semisimple Lie algebras. We classify $\Gamma$-colored $d$-complete posets which specifies the structure of the associated representations. We show that finite $\Gamma$-colored $d$-complete posets are precisely the dominant minuscule heaps of J.R. Stembridge. These heaps are reformulations and extensions of the colored $d$-complete posets of R.A. Proctor. We also show that connected infinite $\Gamma$-colored $d$-complete posets are precisely order filters of the connected full heaps of R.M. Green.