novel applications of algebra and analysis, along with new computational techniques, to determine exact and asymptotic enumeration formulas. One key predictor of the nature of a model’s generating function (whether it is rational, algebraic, or transcendental D-finite, or none of these) is the order of a group that is associated to each model. This group has its origins in the probabilistic study of random walks, namely [@FaIaMa99], and when the group is finite it can sometimes be used to write generating functions as the positive part of an explicit multivariate rational Laurent series. The intimate relation between the generating function of the walks and the nature of the generating function is explored in [@BoRaSa14; @MeMi14a].

For highly symmetric models in two and three dimensions, this group coincides with that of a Weyl group for walks in the Weyl chambers $A_1^2$ and $A_1^3$, respectively. Indeed, one can use either viewpoint to generalize the study of highly symmetric models to models in arbitrary dimension. As these viewpoints are largely isomorphic, and the kernel method viewpoint is more self-contained, we begin this article by working through a straightforward generalization of the kernel method in order to write the generating function for higher dimensional 