Metacommutation of primes in Eichler orders

TL;DR

This paper investigates the metacommutation of primes in Eichler orders, extending known results from maximal orders and providing a combinatorial perspective via Bruhat-Tits trees.

Contribution

It generalizes the cycle structure analysis of metacommutation permutations to Eichler orders and introduces a new combinatorial description using Bruhat-Tits trees.

Findings

Extended cycle structure results to Eichler orders

Provided a combinatorial model using Bruhat-Tits trees

Connected algebraic properties with geometric representations

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.