D. Sagaki, Introduction to LS paths I-II,

I'd like to give survey lectures on Littelmann's path models: In the papers "A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras (1994)" and "Paths and root operators in representation theory (1995)", Littelmann introduced Lakshmibai-Seshadri (LS) paths of shape λ (where λ is an integral weight), and gave the set of them a crystal structure in terms of his root operators; if λ is dominant, then LS paths of shape λ are defined in terms of the Bruhat order (or, Bruhat graph), and the crystal of LS paths of shape λ is isomorphic to the crystal basis of the integrable highest weight module of highest weight λ (proved independently by Kashiwara and Joseph). Littelmann also gave a characterization of the Demazure subcrystal (which is contained in the crystal basis of an integrable highest weight module) in terms of the initial directions of LS paths.

If I have time, I'll also explain quantum LS paths and semi-infinite LS paths, which were introduced in my joint work with Lenart, Naito, Schilling, Shimozono, and joint work with Ishii and Naito, respectively. Quantum (resp., semi-infinite) LS paths are defined in terms of the quantum (resp., semi-infinite) Bruhat graph, instead of the ordinary Bruhat graph, and the crystal of them is isomorphic to the crystal basis of a Weyl module, i.e., a tensor product of level-zero fundamental modules (resp., an extremal weight module).