The author has developed the path model for symmetrizable Kac-Moody algebras in recent papers [ P. Littelmann, Ann. Math., II. Ser. 142, 499-525 (1995; Zbl 0858.17023), Proc. Int. Congr. Math., Zürich 1994, Vol. 1, 298-308 (1995; Zbl 0848.17021)]. The path model is a combinatorial tool for the study of highest weight modules for such Lie algebras, and can be regarded as a generalization of the classical Young tableaux theory for \(SL_n({\mathbb C})\).
The aim of this paper is to give an introduction to the path model. For simplicity, it is restricted to the case of semisimple Lie algebras. This allows the author to give direct proofs of the most important statements in the paper, which are different and simpler from those in the papers mentioned above.
Let \(\mathfrak g\) be a complex semisimple Lie algebra. The author considers the set \(\Pi\) of piecewise linear paths in \({\mathfrak H}_\mathbb R^*\), the real form of the weight lattice of \(\mathfrak g\), which start at the origin and finish at a weight. For each simple root \(\alpha\), he defines operators \(e_{\alpha}\) and \(f_{\alpha}\) on such paths. Let \(\lambda\) be a dominant weight of \(\mathfrak g\), and \(\pi\) a path in \(\Pi\) whose image is contained in the dominant Weyl chamber and which ends at \(\lambda\). Let \({\mathbb B}_{\pi}\) be the set of paths generated from \(\pi\) by applying the root operators. The author shows that \(\text{Char}({\mathbb{B}}_{\pi}):= \sum_{\eta\in {\mathbb{B}}_{\pi}}e^{\eta(1)}\) is equal to the character of the module for \({\mathfrak g}\) with highest weight \(\lambda\). He also derives a generalized Littlewood-Richardson rule which describes how the tensor product of two such modules decomposes into a direct sum of such modules in terms of paths, and also a Demazure character formula, as well as an alternative proof of the P-R-V Conjecture.
For the entire collection see [Zbl 0878.00050].