Gr\"obner-Shirshov bases for Coxeter groups I

TL;DR

This paper investigates the Gr"obner-Shirshov bases for Coxeter groups, providing counterexamples to a previous conjecture and identifying specific cases where the bases can be explicitly constructed.

Contribution

It disproves a general conjecture about Gr"obner-Shirshov bases for Coxeter groups and offers explicit bases for certain cases.

Findings

Counterexamples to the conjecture are provided.

All nontrivial inclusion compositions are classified.

Explicit Gr"obner-Shirshov bases are constructed for specific Coxeter groups.

Abstract

A conjecture of Gr\"obner-Shirshov basis of any Coxeter group has proposed by L.A. Bokut and L.-S. Shiao \cite{bs01}. In this paper, we give an example to show that the conjecture is not true in general. We list all possible nontrivial inclusion compositions when we deal with the general cases of the Coxeter groups. We give a Gr\"obner-Shirshov basis of a Coxeter group which is without nontrivial inclusion compositions mentioned the above.