Chain-independence conjecture for admissible-subset crystal graphs

Let $\mathfrak g$ be a complex symmetrizable Kac–Moody algebra, let $\lambda$ be a dominant weight, and let a $\lambda$-chain be any chain used to define admissible subsets and the associated root operators. These root operators define a colored directed graph on the admissible subsets.

Chain-independence conjecture. The colored directed graph defined by the action of root operators on the admissible subsets corresponding to any $\lambda$-chain does not depend on the choice of this chain.

This conjecture asserts that the combinatorial crystal graph is intrinsic to the weight and the underlying Kac–Moody algebra, rather than to the chosen $\lambda$-chain. The preceding corollary establishes the corresponding crystal-graph identification for a special choice of $\lambda$-chain; the independence for arbitrary $\lambda$-chains is proposed as an analogue of a result of Littelmann.