On the combinatorics of crystal graphs, I. Lusztig's involution

In this paper, we continue the development of a new combinatorial model for the irreducible characters of a complex semisimple Lie group. This model, which will be referred to as the alcove path model, can be viewed as a discrete counterpart to the Littelmann path model. It leads to an extensive generalization of the combinatorics of irreducible characters from Lie type A (where the combinatorics is based on Young tableaux, for instance) to arbitrary type; our approach is type-independent. The main results of this paper are: (1) a combinatorial description of the crystal graphs corresponding to the irreducible representations (this result includes a transparent proof, based on the Yang–Baxter equation, of the fact that the mentioned description does not depend on the choice involved in our model); (2) a combinatorial realization (which is the first direct generalization of Schützenberger's involution on tableaux) of Lusztig's involution on the canonical basis exhibiting the crystals as self-dual posets; (3) an analog for arbitrary root systems, based on the Yang–Baxter equation, of Schützenberger's sliding algorithm, which is also known as jeu de taquin (this algorithm has many applications to the representation theory of the Lie algebra of type A).