Simple Explanation And/or Detailed Examples For Enuemrating Reduced Words Of Length N In Coxeter Groups
Introduction
Coxeter groups are a fundamental concept in geometric group theory, and their study has far-reaching implications in various areas of mathematics. In this article, we will delve into the world of Coxeter groups and explore the problem of enumerating reduced words of length n. We will provide a simple explanation and detailed examples to help readers understand this complex topic.
What are Coxeter Groups?
A Coxeter group is a group generated by a set of involutions (elements of order 2) subject to certain relations. These relations are specified by a Coxeter matrix, which encodes the information about the generators and their interactions. Coxeter groups are named after the mathematician H.S.M. Coxeter, who first introduced them in the 1930s.
Reduced Words in Coxeter Groups
In a Coxeter group, a reduced word is a sequence of generators that represents an element of the group. The length of a reduced word is the number of generators in the sequence. Reduced words play a crucial role in the study of Coxeter groups, as they provide a way to describe the group's structure and properties.
Enumerating Reduced Words of Length n
The problem of enumerating reduced words of length n in a Coxeter group is a fundamental question in the field. It involves counting the number of reduced words of length n that can be formed using the generators of the group. This problem has important implications in various areas of mathematics, including combinatorics, algebra, and geometry.
The Coxeter Matrix and its Role in Enumerating Reduced Words
The Coxeter matrix plays a central role in enumerating reduced words of length n. It encodes the information about the generators and their interactions, which is essential for counting the number of reduced words. The Coxeter matrix is a square matrix with entries that are either 1, 2, or infinity. The entries of the matrix specify the relations between the generators, and they determine the structure of the Coxeter group.
Examples of Coxeter Groups and their Reduced Words
Let's consider some examples of Coxeter groups and their reduced words. We will use the Coxeter matrix to enumerate the reduced words of length n.
Example 1: The Symmetric Group S3
The symmetric group S3 is a Coxeter group with 3 generators. The Coxeter matrix is:
| 1 | 2 | 3 | |
|---|---|---|---|
| 1 | 1 | 2 | 2 |
| 2 | 2 | 1 | 2 |
| 3 | 2 | 2 | 1 |
The reduced words of length n in S3 can be enumerated using the Coxeter matrix. For example, the reduced words of length 2 are:
- (1, 2)
- (1, 3)
- (2, 1)
- (2, 3)
- (3, 1)
- (3, 2)
Example 2: The Coxeter Group G(4, 3)
The Coxeter group G(4, 3) is a Coxeter group with 4 generators. The Coxeter matrix is:
| 2 | 3 | 4 | ||
|---|---|---|---|---|
| 1 | 1 | 2 | 2 | 2 |
| 2 | 2 | 1 | 2 | 2 |
| 3 | 2 | 2 | 1 | 2 |
| 4 | 2 | 2 | 2 | 1 |
The reduced words of length n in G(4, 3) can be enumerated using the Coxeter matrix. For example, the reduced words of length 2 are:
- (1, 2)
- (1, 3)
- (1, 4)
- (2, 1)
- (2, 3)
- (2, 4)
- (3, 1)
- (3, 2)
- (3, 4)
- (4, 1)
- (4, 2)
- (4, 3)
Conclusion
In this article, we have provided a simple explanation and detailed examples of enumerating reduced words of length n in Coxeter groups. We have introduced the concept of Coxeter groups and their reduced words, and we have used the Coxeter matrix to enumerate the reduced words of length n in two examples. The Coxeter matrix plays a central role in enumerating reduced words, and it encodes the information about the generators and their interactions. We hope that this article has provided a useful introduction to this complex topic and has inspired readers to explore the world of Coxeter groups.
References
- Coxeter, H.S.M. (1934). "Discrete groups generated by reflections." Annals of Mathematics, 35(2), 588-621.
- Humphreys, J.E. (1990). Reflection Groups and Coxeter Groups. Cambridge University Press.
- Björner, A., and Brenti, F. (2005). Combinatorics of Coxeter Groups. Springer-Verlag.
Further Reading
- The OEIS (On-Line Encyclopedia of Integer Sequences) is a valuable resource for exploring the properties of Coxeter groups and their reduced words.
- The book "Combinatorics of Coxeter Groups" by Anders Björner and Francesco Brenti provides a comprehensive introduction to the combinatorial properties of Coxeter groups.
- The article "Discrete groups generated by reflections" by H.S.M. Coxeter provides a classic introduction to the concept of Coxeter groups and their reduced words.
Q&A: Enumerating Reduced Words of Length n in Coxeter Groups ===========================================================
Introduction
In our previous article, we explored the concept of Coxeter groups and their reduced words. We introduced the Coxeter matrix and used it to enumerate the reduced words of length n in two examples. In this article, we will answer some frequently asked questions about enumerating reduced words of length n in Coxeter groups.
Q: What is the significance of the Coxeter matrix in enumerating reduced words?
A: The Coxeter matrix plays a central role in enumerating reduced words of length n in Coxeter groups. It encodes the information about the generators and their interactions, which is essential for counting the number of reduced words.
Q: How do I determine the Coxeter matrix for a given Coxeter group?
A: The Coxeter matrix can be determined from the presentation of the Coxeter group. The presentation specifies the generators and their relations, which can be used to construct the Coxeter matrix.
Q: What is the difference between a reduced word and a non-reduced word in a Coxeter group?
A: A reduced word is a sequence of generators that represents an element of the group, while a non-reduced word is a sequence of generators that can be simplified to a reduced word.
Q: How do I enumerate the reduced words of length n in a Coxeter group?
A: To enumerate the reduced words of length n in a Coxeter group, you can use the Coxeter matrix to count the number of reduced words. This involves using the matrix to determine the number of possible sequences of generators of length n.
Q: What is the relationship between the Coxeter matrix and the Bruhat order?
A: The Coxeter matrix is closely related to the Bruhat order, which is a partial order on the elements of the Coxeter group. The Bruhat order can be used to determine the number of reduced words of length n in the group.
Q: Can I use the Coxeter matrix to enumerate the reduced words of length n in a non-Coxeter group?
A: No, the Coxeter matrix is specific to Coxeter groups and cannot be used to enumerate the reduced words of length n in a non-Coxeter group.
Q: Are there any algorithms for enumerating reduced words of length n in Coxeter groups?
A: Yes, there are several algorithms for enumerating reduced words of length n in Coxeter groups. These algorithms can be used to count the number of reduced words and to determine the properties of the group.
Q: Can I use the Coxeter matrix to determine the properties of a Coxeter group?
A: Yes, the Coxeter matrix can be used to determine the properties of a Coxeter group, such as the order of the group and the structure of the group's lattice.
Q: Are there any software packages or libraries that can be used to enumerate reduced words of length n in Coxeter groups?
A: Yes, there are several software packages and libraries that can be used to enumerate reduced words of length n in Coxeter groups. These include Sage, GAP, and Mathematica.
Conclusion
In this article, we have answered some frequently asked questions about enumerating reduced words of length n in Coxeter groups. We have discussed the significance of the Coxeter matrix, how to determine the Coxeter matrix, and how to enumerate the reduced words of length n. We have also discussed the relationship between the Coxeter matrix and the Bruhat order, and the use of algorithms and software packages to enumerate reduced words.
References
- Coxeter, H.S.M. (1934). "Discrete groups generated by reflections." Annals of Mathematics, 35(2), 588-621.
- Humphreys, J.E. (1990). Reflection Groups and Coxeter Groups. Cambridge University Press.
- Björner, A., and Brenti, F. (2005). Combinatorics of Coxeter Groups. Springer-Verlag.
Further Reading
- The OEIS (On-Line Encyclopedia of Integer Sequences) is a valuable resource for exploring the properties of Coxeter groups and their reduced words.
- The book "Combinatorics of Coxeter Groups" by Anders Björner and Francesco Brenti provides a comprehensive introduction to the combinatorial properties of Coxeter groups.
- The article "Discrete groups generated by reflections" by H.S.M. Coxeter provides a classic introduction to the concept of Coxeter groups and their reduced words.