Let \({\mathfrak g}\) be a symmetrizable Kac-Moody algebra and \(X\) the weight lattice of \({\mathfrak g}\). Let \(\Pi\) denote the set of all piecewise linear paths \(\pi : [0, 1]_\mathbb{Q} \to X_\mathbb{Q}\) such that \(\pi (0) = O\) and \(\pi (1) =\) an integral weight. For each simple root \(\alpha\), the author defines the path operators \(e_\alpha\) and \(f_\alpha\). These operators have many interesting properties which are similar to those of Kashiwara’s operators in the crystal basis theory [M. Kashiwara, Duke Math. J. 63, 465-516 (1991; Zbl 0739.17005)]. For example, the concatenation of paths behaves similarly to the tensor product of modules over \({\mathfrak g}\), and the path operators behave nicely with respect to the concatenation as Kashiwara’s operators do with respect to the tensor product. Let \({\mathcal A}\) be the algebra generated by the path operators and \(\Pi^+\) be the set of all paths whose images are completely contained in the dominant Weyl chamber. Then the integral module \({\mathcal A} \pi\) is interpreted as a “model” for the irreducible highest weight module \(V (\lambda)\) over \({\mathfrak g}\) where \(\lambda = \pi (1)\). In particular, we have \(\text{Char} {\mathcal A} \pi = \text{Char} V (\lambda)\). Moreover, the author shows that the concatenation of \({\mathcal A}\)-modules can be viewed as the tensor product of modules over \({\mathfrak g}\), and gives a decomposition rule for the tensor products which is a generalization of the Littlewood-Richardson rule. As a by-product, the author gives a purely combinatorial proof of the PRV conjecture.
For the entire collection see [Zbl 0829.00014].