Littelmann path model for crystals (awaiting review)

The Littelmann path model provides a unified combinatorial model for highest weight crystals that applies to all symmetrizable Kac-Moody algebras.

Let $P$ be the weight lattice in the dual of a Cartan subalgebra of the semisimple Lie algebra $\mathfrak{g}$. A Littelmann path is a piecewise-linear mapping \[ \pi:[0,1]\cap \mathbf{Q} \rightarrow P\otimes_{\mathbf{Z}}\mathbf{Q} \] such that $\pi(0) = 0$ and $\pi(1)$ are weights. Littelmann defined the Kashiwara raising and lowering operators $e_i$ and $f_i$ on these paths.