Powers of Coxeter elements with unbounded reflection length

TL;DR

This paper investigates the behavior of powers of Coxeter elements in certain Coxeter groups, showing they can have unbounded reflection length and establishing bounds in specific cases, thus advancing understanding of their algebraic structure.

Contribution

It introduces new results on the unboundedness of reflection length for Coxeter elements and links reflection length functions across different Coxeter groups.

Findings

Powers of Coxeter elements can have unbounded reflection length in groups with large braid relations.

Established upper bounds for reflection length in specific Coxeter groups.

Connected reflection length functions between arbitrary and universal Coxeter groups.

Abstract

For Coxeter groups with sufficiently large braid relations, we prove that the sequence of powers of a Coxeter element has unbounded reflection length. We establish a connection between the reflection length functions on arbitrary Coxeter groups and the reflection length functions on universal Coxeter groups of the same rank through the solution to the word problem for Coxeter groups. For Coxeter groups corresponding to a Coxeter matrix with the same entry everywhere except the diagonal, upper bounds for the reflection length of the powers of Coxeter elements are established.