On the connection between Young walls and Littelmann paths

In the article Jung et al. (2009) [6], , we developed the combinatorics of Young walls associated with higher level adjoint crystals for the quantum affine algebra . The irreducible highest weight crystal B(λ) of arbitrary level is realized as the affine crystal consisting of reduced Young walls on λ. Littelmann (1994) [16], gave a concrete path model with which Joseph and Kashiwara proved that there exists an isomorphism between the crystal graph and the Littelmann paths graph. In this article, using the Young wall constructed in Jung et al. (2009) [6] we try to give in combinatorial way the bijective correspondence between the affine crystal consisting of reduced Young walls on Λ0 and the Littelmann path model.