Classifications of \(\Gamma\)-colored \(d\)-complete posets and upper \(P\)-minuscule Borel representations (Q2223468)

Summary: 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 \textit{J. R. Stembridge} [J. Algebra 235, No. 2, 722--743 (2001; Zbl 0973.17034)]. These heaps are reformulations and extensions of the colored \(d\)-complete posets of \textit{R. A. Proctor} [J. Algebra 213, No. 1, 272--303 (1999; Zbl 0969.05068)]. We also show that connected infinite \(\Gamma\)-colored \(d\)-complete posets are precisely order filters of the connected full heaps of \textit{R. M. Green} [Combinatorics of minuscule representations. Cambridge: Cambridge University Press (2013; Zbl 1320.17005)].