A Combinatorial Model for Crystals of Ka

We present a simple combinatorial model for the characters of the irreducible integrable highest weight modules for complex symmetrizable Kac-Moody algebras. This model can be viewed as a discrete counterpart to the Littelmann path model. We describe crystal graphs and propose a Littlewood-Richardson rule for decomposing tensor products of irreducible representations. The foundation of this new model is based on the concept of a λ-chain, which is defined as a chain of positive roots constrained by certain interlacing conditions. Our research builds upon previous work in the field, particularly concerning irreducible characters of complex semisimple Lie groups and Demazure characters, providing novel insights into the structure and enumeration of saturated chains in the Bruhat order on the respective Weyl group. By utilizing alcove paths in the context of the affine Weyl group, we articulate connection to decompositions of elements within this framework, thus enriching the combinatorial understanding of Kac-Moody representations. The results derived from this model enhance the existing knowledge surrounding character formulas expressed in terms of R-matrices, governed by the Yang-Baxter equation.