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Combinatorial group and representation theory?

by pivoxa15
Tags: combinatorial, representation, theory
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pivoxa15
#1
Dec8-07, 02:19 PM
P: 2,267
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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Hurkyl
#2
Dec8-07, 02:35 PM
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PF Gold
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I would imagine that combinatorial group theory is the study of groups via combinatorial methods.
Chris Hillman
#3
Dec8-07, 03:21 PM
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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.

pivoxa15
#4
Dec8-07, 05:53 PM
P: 2,267
Combinatorial group and representation theory?

Thanks Chris. What about the question of the connection between Combinatorial group theory and (combinatorial) representation theory?


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