Linearity of Generalized Cactus Groups

TL;DR

This paper generalizes the embedding of pure cactus groups into right-angled Coxeter groups to cactus groups associated with arbitrary finite Coxeter groups, demonstrating their linearity through new representations.

Contribution

It extends the known embedding and linearity results from symmetric groups to all finite Coxeter groups, broadening the understanding of cactus groups.

Findings

Pure cactus groups embed into right-angled Coxeter groups.

Generalized cactus groups are linear.

New representations support linearity proof.

Abstract

Cactus groups are traditionally defined based on symmetric groups, and pure cactus groups are particular subgroups of cactus groups. Mostovoy showed that pure cactus groups embed into right-angled Coxeter groups. We generalize this result to cactus groups associated with arbitrary finite Coxeter groups and we investigate some representations of generalized cactus groups and deduce the linearity of generalized cactus groups.