On the combinatorics of crystal graphs

In this paper, we continue the development of a new combinatorial model for the irreducible characters of a complex semisimple Lie group, referred to as the alcove path model. This model represents a discrete counterpart to the Littelmann path model and leads to an extensive generalization of the combinatorics of irreducible characters across various Lie types, extending beyond the classical type A. The main results include: a combinatorial description of the crystal graphs corresponding to irreducible representations, bolstered by a proof based on the Yang-Baxter equation; a combinatorial realization of an important involution on the canonical basis illustrating the self-duality of crystals and their connection to the longest Weyl group element's action; and an analog for arbitrary root systems regarding Schützenberger’s sliding algorithm, known as jeu de taquin, demonstrating numerous applications in the representation theory of the Lie algebra of type A. We thus invite further exploration and application of this model in the broader context of representation theory.