In Littelmann’s Inventiones Mathematicae paper A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras, the notion of a Laskshmibai-Seshadri path or LS path was defined. The basic idea is to generalise the notion of semistandard Young tableaux into any symmetrisable Kac-Moody type by replacing the tableau by a piecewise-linear path in the weight lattice, starting at 0 and ending at a weight. Suitable e_i and f_i operators can be defined on the set of paths, giving it the structure of a crystal.

A path in this context is a piecewise-linear path \pi \colon [0, 1] \to X_\bbR in the weight space, starting at the origin (\pi(0) = 0), with two paths being considered equal if one is a reparametrisation of the other. Each root operator acts by cutting the path into three pieces \pi = \pi_1 * \pi_2 * \pi_3, where * is a suitable concatenation of paths-starting-at-the-origin. The root operator either kills the path, or returns \pi_1 * s_\alpha(\pi_2) * \pi_3.

For a dominant weight \lambda, let \pi_\lambda be the path connecting the origin to \lambda in a straight line. The set of LS paths of shape \lambda are as the set of paths generated by repeatedly acting on \pi_\lambda by the f_\alpha. This set of paths under the e_\alpha and f_\alpha operators form a crystal, and so we get a whole host of amazing consequences for the combinatorial information about V(\lambda):

1. The weight of an LS path \pi is its ending point \pi(1). The dimension of the V(\lambda)_\mu weight space is the number of LS paths of shape \lambda of weight \mu.
2. Call a path \pi of weight \mu dominant if the translated image \mu + \pi([0, 1]) is contained in the dominant chamber (\mu being a dominant weight is necessary, but not sufficient, for \pi to be dominant). Then the multiplicities of irreducibles in the tensor product V(\lambda) \otimes V(\mu) are given by the weights \lambda + \pi(1), where \pi runs over all \mu-dominant LS paths of shape \lambda.