Séminaire Lotharingien de Combinatoire, 95B.18 (2026), 12 pp.
Andrew Elvey Price, Emmanuel Humbert and Kilian Raschel
On the Finiteness of the Group Associated with Weighted Walks in multidimensional Orthants
Abstract.
In the study of walks with small steps confined to multidimensional orthants, a certain group of transformations plays a central role. In particular, several techniques to potentially compute the generating function, including the orbit sum method, can only be applied when this group is finite. In this extended abstract, we present three new results concerning this group. First, in two dimensions, we provide a complete characterization of the weight parameters that yield a finite group. In higher dimensions, we show that whenever the group is finite, it must necessarily be isomorphic to a simpler reflection group. Finally, in three dimensions, we give a full classification of the parameters leading to a finite group that also satisfies an additional Weyl property.
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
The following versions are available: