Cactus groups and Lusztig's asymptotic algebra

TL;DR

This paper establishes a connection between cactus groups and Lusztig's asymptotic algebra, proposing a conjecture to recover known Coxeter group actions, advancing understanding of algebraic symmetries.

Contribution

It constructs a morphism linking cactus groups to Lusztig's asymptotic algebra and proposes a conjecture to recover Coxeter group actions.

Findings

Constructed a morphism from cactus group to Lusztig's algebra

Related cactus group actions to Coxeter group elements

Proposed a conjecture for full recovery of actions

Abstract

We construct a morphism from the cactus group associated with a Coxeter group to the group of invertible elements of Lusztig's asymptotic algebra. This relates to the cactus group action on elements of Coxeter groups defined by Losev and Bonnaf\'e and we propose a conjecture on how to fully recover those actions.