  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. 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 of Lusztig's involution on the
canonical basis (this involution exhibits the crystals as self-dual posets,
corresponds to the action of the longest Weyl group element on an irreducible
representation, and generalizes Schutzenberger's involution on tableaux); (3)
an analog for arbitrary root systems, based on the Yang-Baxter equation, of
Schutzenberger'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).
