The lowest two-sided cell of weighted Coxeter groups of rank 3

TL;DR

This paper provides a detailed description of the lowest two-sided cell in weighted Coxeter groups of rank 3, verifies several conjectures for this cell, and explores its algebraic structure.

Contribution

It offers the first precise characterization of the lowest two-sided cell in rank 3 weighted Coxeter groups and confirms key conjectures related to its properties.

Findings

Confirmed conjectures P1-P15 and  for c_{0}

Described the structure of the based ring of c_{0}

Provided explicit calculations for the cell's algebraic properties

Abstract

In this paper, we give precise description for the lowest lowest two-sided cell c_{0} and the left cells in it for a weighted Coxeter group of rank 3. Then we show conjectures P1-P15 and \widetilde{P} hold for c_{0} and do some calculation for the based ring of c_{0}.