Electron. J. Comb. | 2021
Classifications of $\\Gamma$-Colored $d$-Complete Posets and Upper $P$-Minuscule Borel 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.