Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Gröbner-Shirshov bases for the Coxeter groups of the types $G_2,F_4,E_6,E_7$ and $E_8$

Xiaowei Pang, Jun Wang

Abstract

The author is mainly interest in the Gröbner-Shirshov bases of finite Coxeter groups. It is known that the finite Coxeter groups are classified in terms of Coxeter-Dynkin diagrams. Under the fixed order, it is worth mention that the presentation of group determines the Gröbner-Shirshov bases of group. In this paper, the author rearranges the generators, marks them on Coxeter-Dynkin diagrams, and gets a simple presentation of the Gröbner-Shirshov bases for Coxeter groups of types $G_2, F_4, E_6$ and $E_7$. This article also gives the Gröbner-Shirshov basis of Coxeter group of type $E_8$.

Gröbner-Shirshov bases for the Coxeter groups of the types $G_2,F_4,E_6,E_7$ and $E_8$

Abstract

The author is mainly interest in the Gröbner-Shirshov bases of finite Coxeter groups. It is known that the finite Coxeter groups are classified in terms of Coxeter-Dynkin diagrams. Under the fixed order, it is worth mention that the presentation of group determines the Gröbner-Shirshov bases of group. In this paper, the author rearranges the generators, marks them on Coxeter-Dynkin diagrams, and gets a simple presentation of the Gröbner-Shirshov bases for Coxeter groups of types and . This article also gives the Gröbner-Shirshov basis of Coxeter group of type .

Paper Structure

This paper contains 4 sections, 5 theorems, 23 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $Gr\ddot{o}bner-Shirshov$ bases of the Coxeter groups of the types $E_6,E_7,E_8$
  4. $Gr\ddot{o}bner-Shirshov$ bases of the Coxeter groups of the type $G_2,F_4$

Key Result

Theorem 3.2

The Gröbner-Shirshov basis $GB(E_6)$ of the Coxeter group of the type $E_6$ is the set of the following relations together with the initial relations in Definition initial:

Theorems & Definitions (21)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 11 more