Simply-laced mixed-sign Coxeter groups, with an associate graph is a line or a simple cycle

TL;DR

This paper classifies simply-laced mixed-sign Coxeter groups with associated graphs that are lines or simple cycles, revealing their defining relations are squares or cubes of conjugate products linked to vertex labels.

Contribution

It provides a complete classification of these Coxeter groups and describes their relations in terms of vertex labels and conjugate products.

Findings

All relations are squares or cubes of conjugate products.

Relations are strongly connected to vertex labels.

Classification applies to graphs that are lines or simple cycles.

Abstract

In 2011 Eriko Hironaka introduced an interesting generalization of Coxeter groups, motivated by studying certain mapping classes. The generalization is by labeling the vertices of a Coxeter graph either by +1 or by -1, and then generalizing the standard geometric representation of the associated Coxeter group by concerning the labels of the vertices. The group which Hironaka get by that generalization is called mixed-sign Coxeter group. In this paper we classify the simply-laced mixed-sign Coxeter groups where the associated graph is either a line or a simple cycle. We show that all the defining relations of the mixed-sign Coxeter groups with the mentioned associated graph (either a line or a simple cycle) are squares or cubes of a product of conjugates of two generators of the mixed-sign Coxeter group and are strongly connected to the labels of the vertices of the associated graph.