Metacommutation of primes in Eichler orders

Abstract

In this article, we study the metacommutation problem in locally Eichler orders. From this arises a permutation of the set of locally principal left ideals of a given prime reduced norm. Previous results on the cycle structure were determined for locally maximal orders. As we extend these results, we present an alternative, combinatorial description of the metacommutation permutation as an action on the Bruhat-Tits tree.