Connectivity Properties of Factorization Posets in Generated Groups

TL;DR

This paper explores different notions of connectivity in posets derived from group factorizations, introducing a unified framework and tools to analyze their interactions and properties.

Contribution

It introduces a new framework for studying connectivity properties in factorization posets using compatible linear orders of generators.

Findings

Established connections between different connectivity notions

Developed a unified approach to analyze these properties

Provided tools for studying factorization posets in generated groups

Abstract

We consider three notions of connectivity and their interactions in partially ordered sets coming from reduced factorizations of an element in a generated group. While one form of connectivity essentially reflects the connectivity of the poset diagram, the other two are a bit more involved: Hurwitz-connectivity has its origins in algebraic geometry, and shellability in topology. We propose a framework to study these connectivity properties in a uniform way. Our main tool is a certain linear order of the generators that is compatible with the chosen element.