Relation between the Weyl group orbits of fundamental weights for multiply-laced finite dimensional simple Lie algebras and d-complete posets

TL;DR

This paper explores the relationship between Weyl group orbits of fundamental weights and d-complete posets in multiply-laced simple Lie algebras, extending known results from simply-laced cases using diagram automorphisms.

Contribution

It extends the known correspondence between Weyl group orbits and d-complete posets from simply-laced to multiply-laced Lie algebras via folding techniques.

Findings

Established a connection between Weyl group orbits and d-complete posets for multiply-laced Lie algebras.

Generalized the order isomorphism using Dynkin diagram automorphisms.

Enhanced understanding of the combinatorial structure of Lie algebra representations.

Abstract

It is known that there exists an order isomorphism between the Weyl group orbit through a minuscule weight of a simply-laced finite-dimensional simple Lie algebra and the set of all order filters in a self-dual connected d-complete poset. In this paper, we try to extend this fact to the case of multiply-laced finite-dimensional simple Lie algebras by using the "folding" technique with respect to a Dynkin diagram automorphism.