Where Does The Convention For The Letter W W W For Coxeter Groups Come From?

Introduction

In the realm of abstract algebra, particularly in the study of Coxeter groups, a specific convention has been adopted to denote these groups using the letter $W$. This convention may seem arbitrary at first, but it has a rich history and significance in the field. In this article, we will delve into the origins of this convention and explore the reasons behind the choice of the letter $W$ for Coxeter groups.

A Brief History of Coxeter Groups

Coxeter groups are a class of groups that were first introduced by Harold Scott MacDonald Coxeter in the 1930s. These groups are characterized by their presentation, which involves a set of generators and relations. The presentation of a Coxeter group is typically denoted by a matrix, known as the Coxeter matrix, which encodes the relations between the generators. Coxeter groups have since become a fundamental area of study in abstract algebra, with applications in geometry, combinatorics, and other areas of mathematics.

The Convention of Using $W$

Q: What is the significance of the letter $W$ in Coxeter groups?

A: The letter $W$ is used to denote the Weyl group of a Coxeter system, which is a fundamental concept in the study of Coxeter groups. The Weyl group is a group of symmetries of the Coxeter graph, and it plays a crucial role in the classification of Coxeter groups.

Q: Why is the Weyl group important in Coxeter groups?

A: The Weyl group is important because it provides a way to classify Coxeter groups based on their Coxeter graph. The Weyl group is a group of symmetries of the Coxeter graph, and it can be used to determine the structure of the Coxeter group. In particular, the Weyl group can be used to determine the order of the Coxeter group, as well as the conjugacy classes of its elements.

Q: What is the relationship between the Weyl group and the Coxeter matrix?

A: The Weyl group is closely related to the Coxeter matrix, which is a matrix that encodes the relations between the generators of the Coxeter group. The Weyl group can be used to determine the Coxeter matrix, and vice versa. In particular, the Weyl group can be used to determine the order of the Coxeter matrix, as well as the conjugacy classes of its elements.

Q: How did the convention of using $W$ for the Weyl group originate?

A: The convention of using $W$ for the Weyl group originated from the work of Coxeter himself. In his 1934 paper, Coxeter introduced the concept of the Weyl group and used the letter $W$ to denote it. This convention has since been adopted by the mathematical community and is now widely used in the study of Coxeter groups.

Q: Are there any other conventions for denoting the Weyl group?

A: While the convention of using $W$ for the Weyl group is widely accepted, there are other conventions that have been used in the past. For example, some authors have used the letter $G$ to denote the Weyl group, while others have used the letter $H$. However, the convention of using $W$ is now the most widely accepted and is used in most mathematical texts.

Q: What are some of the key properties of the Weyl group?

A: The Weyl group has several key properties that make it an important object of study in the theory of Coxeter groups. Some of the key properties of the Weyl group include:

• The Weyl group is a group of symmetries of the Coxeter graph.
• The Weyl group can be used to determine the structure of the Coxeter group.
• The Weyl group can be used to determine the order of the Coxeter group.
• The Weyl group can be used to determine the conjugacy classes of the elements of the Coxeter group.

Q: How is the Weyl group used in the study of Coxeter groups?

A: The Weyl group is a fundamental object of study in the theory of Coxeter groups. It is used to determine the structure of the Coxeter group, as well as the conjugacy classes of its elements. The Weyl group is also used to classify Coxeter groups based on their Coxeter graph. In particular, the Weyl group can be used to determine the order of the Coxeter group, as well as the conjugacy classes of its elements.

Q: What are some of the applications of the Weyl group in mathematics?

A: The Weyl group has several applications in mathematics, including:

• The study of Coxeter groups and their properties.
• The classification of Coxeter groups based on their Coxeter graph.
• The determination of the order of the Coxeter group.
• The determination of the conjugacy classes of the elements of the Coxeter group.
• The study of the symmetries of the Coxeter graph.

Q: What are some of the open problems in the study of the Weyl group?

A: There are several open problems in the study of the Weyl group, including:

• The determination of the structure of the Weyl group for arbitrary Coxeter graphs.
• The classification of Coxeter groups based on their Coxeter graph.
• The determination of the order of the Coxeter group.
• The determination of the conjugacy classes of the elements of the Coxeter group.
• The study of the symmetries of the Coxeter graph.