Generalisations of the Tits representation

TL;DR

This paper introduces a new group K_n with properties akin to infinite Coxeter groups, featuring a geometric representation with hyperplanes and chambers, and explores its combinatorial properties.

Contribution

It constructs the group K_n with a geometric representation and analyzes its combinatorial structure, extending Coxeter group concepts.

Findings

K_n has a geometric representation with hyperplanes and chambers

Finite residues of K_n exhibit specific combinatorial properties

Generators are based on 2-element subsets of {0, ..., n}

Abstract

We construct a group K_n with properties similar to infinite Coxeter groups. In particular, it has a geometric representation featuring hyperplanes and simplicial chambers. The generators of K_n are given by 2-element subsets of {0, .., n}. We give some easy combinatorial results on the finite residues of K_n.