Symmetric Presentations of Coxeter Groups

TL;DR

This paper uses symmetric generation techniques to derive standard presentations and explicit representations of finite simply laced Coxeter groups of types A, D, and E, highlighting their natural connection to symmetric group actions.

Contribution

It introduces a novel approach to obtaining Coxeter group presentations and representations via symmetric generation and natural symmetric group actions.

Findings

Derived standard presentations for types A, D, E Coxeter groups.

Provided explicit group representations.

Connected Coxeter groups to symmetric group actions.

Abstract

We apply the techniques of symmetric generation to establish the standard presentations of the finite simply laced irreducible finite Coxeter groups, that is the Coxeter groups of types An, Dn and En, and show that these are naturally arrived at purely through consideration of certain natural actions of symmetric groups. We go on to use these techniques to provide explicit representations of these groups.