Combinatorial group and representation theory?

In summary: Representation theory is a branch of mathematics that studies the properties of mathematical objects that can be described by some finite set of linear equations. In particular, it is concerned with the ways that these objects can be mapped into each other. This connection is made through the concept of a representation. A representation is a way of representing an object in terms of other objects. In other words, it is a mapping from an object to another object. The most common type of representation is a mapping from an object to a set of points in space. For example, the number 3 can be represented as a set of points in 3-dimensional space. Another type of representation is a mapping from an object to
  • #1
pivoxa15
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Why is there 'combinatorial' in front of both of combinatorial group theory and combinatorial representation theory? What does it imply?

What is the difference and similiarities between cgt and crt besides the fact that they both have the word combinatorial in front of them?
 
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  • #2
I would imagine that combinatorial group theory is the study of groups via combinatorial methods.
 
  • #3
Suggest some good books

Indeed, and it is a venerable, lovely, and important subject, which important connections with topology, computational group theory. In particular, the theory of reflection groups can be taken as a subdiscipline and this is closely related to topics in Lie algebras, geometric invariant theory, geometric enumerative combinatorics (e.g. Schubert calculus), and more!

This happens to be a fairly well defined field blessed by numerous good book-length introductions to the subject.

An older classic textbook:

Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations, 2nd ed., Dover reprint.

A nice pair of more modern and very readable advanced undergraduate level textbooks:

D. L. Johnson, Presentations of Groups, London Mathematical Society student texts Vol. 15, Cambridge University Press, 1990.

Daniel E. Cohen, Combinatorial Group Theory: a Topological Approach, London Mathematical Society student texts Vol. 14, Cambridge University Press, 1989.

An historical account (by two well known figures in this field):

Bruce Chandler and Wilhelm Magnus, The History of Combinatorial Group Theory: a Case Study in the History of Ideas, Springer-Verlag, 1982.

Some more specialized books:

Roger C. Lyndon and Paul E. Schupp, Combinatorial Group Theory, Springer, 1977.

John Stillwell, Classical Topology and Combinatorial Group Theory, Springer, 1980.

Gilbert Baumslag, Topics in Combinatorial Group Theory, Birkhäuser, 1993.

Richard Kane, Reflection Groups and Invariant Theory, Springer, 2001.

A classic monograph:

H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer, 1980.
 
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  • #4
Thanks Chris. What about the question of the connection between Combinatorial group theory and (combinatorial) representation theory?
 

1. What is combinatorial group theory?

Combinatorial group theory is a branch of abstract algebra that studies the properties of groups using combinatorial techniques. It focuses on the structure and classification of groups, as well as the relationships between different groups.

2. What is representation theory?

Representation theory is a branch of mathematics that studies how abstract algebraic structures, such as groups, rings, and algebras, can be represented as linear transformations of vector spaces. It provides a powerful tool for understanding and analyzing these structures.

3. How are combinatorial group theory and representation theory related?

Combinatorial group theory and representation theory are closely related as representation theory provides a way to study the structure of groups using linear algebraic techniques. It allows for a deeper understanding of group theory and has many applications in other areas of mathematics and physics.

4. What are some applications of combinatorial group and representation theory?

Combinatorial group and representation theory have many practical applications in fields such as cryptography, coding theory, and quantum mechanics. They also have connections to other areas of mathematics, including topology, geometry, and number theory.

5. What are some open problems in combinatorial group and representation theory?

Some open problems in combinatorial group and representation theory include the classification of finite simple groups, the McKay conjecture, and the inverse Galois problem. There is also ongoing research on the relationships between these two theories and their connections to other areas of mathematics.

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