Combinatorial group and representation theory?

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Discussion Overview

The discussion centers on the terminology and implications of "combinatorial" in combinatorial group theory (CGT) and combinatorial representation theory (CRT). Participants explore the differences and similarities between these two fields, as well as their connections to other mathematical areas.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant questions the significance of the term "combinatorial" in both CGT and CRT and seeks to understand the implications of this terminology.
  • Another participant suggests that combinatorial group theory involves studying groups through combinatorial methods.
  • A later reply highlights the connections of CGT with topology and computational group theory, mentioning its relationship with reflection groups, Lie algebras, and geometric enumerative combinatorics.
  • Several books on CGT are recommended, indicating a rich literature on the subject, including classic and modern texts.
  • A participant raises a question about the connection between CGT and CRT, indicating an interest in exploring the relationship between these two areas further.

Areas of Agreement / Disagreement

Participants have not reached a consensus on the implications of the term "combinatorial" or the specific connections between CGT and CRT, indicating that multiple views and uncertainties remain in the discussion.

Contextual Notes

The discussion does not resolve the differences or similarities between CGT and CRT, nor does it clarify the connections between them. The implications of the term "combinatorial" are also left open to interpretation.

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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I would imagine that combinatorial group theory is the study of groups via combinatorial methods.
 
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.
 
Last edited:
Thanks Chris. What about the question of the connection between Combinatorial group theory and (combinatorial) representation theory?
 

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