Understanding Group Representations in Group Theory

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Discussion Overview

The discussion centers on understanding group representations in group theory, specifically the relationship between groups and their representations through matrices. Participants explore concepts such as isomorphism, faithful representations, and the specific case of rotations in three-dimensional space and their connection to the special orthogonal group SO(3).

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant seeks to understand what constitutes a representation of a group.
  • Another participant explains that a representation involves square matrices that reflect the group's multiplication table, with a distinction made for "faithful" representations that are isomorphic.
  • A participant asks for clarification on the term 'isomorphism' and its relation to matrix structure.
  • It is noted that an isomorphism indicates that matrices behave similarly to the group itself.
  • Discussion includes the existence of a trivial representation using the identity matrix, which does not provide meaningful insights about the group.
  • A participant suggests that the group of all rotations in three-dimensional space is isomorphic to SO(3), which is confirmed by another participant.
  • A question is raised regarding the necessity of the determinant being 1 for elements of SO(3) and whether it could be isomorphic to O(3).
  • Another participant responds that the determinant condition relates to distinguishing between proper and improper rotations.
  • A later post questions the difficulty of proving the isomorphism of the group of all rotations in three-dimensional space to SO(3) and draws a parallel to rotations in two dimensions being isomorphic to SO(2).

Areas of Agreement / Disagreement

Participants generally agree on the definitions and properties of group representations and isomorphisms, but there are questions and discussions regarding the specific conditions of determinants and the nature of isomorphisms, indicating some unresolved aspects.

Contextual Notes

Some assumptions about the nature of rotations and the properties of matrices are not fully explored, and the discussion does not resolve the complexities surrounding the isomorphism proofs.

jimmy.neutron
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Hey guys, I'm pretty new to group theory at the moment, what's the best way of understanding a 'representation' of a group?

Thanks
 
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A representation of a group is just a set of square matrices such that matrix multiplication reproduces the group's multiplication table. A "faithful" representation is where the correspondence is an isomorphism.
 
Oh right, that seems to make sense! So finally, what do you mean by an 'isomorphism'? Is this something to do with the structure of the matrices?
 
An isomorphism means that the set of matrices behave the same way as the group.

Many groups have a trivial representation where you represent every group element by the identity matrix. The rules of the group multiplication table are satisfied. But one certainly doesn't learn anything interesting about the group by looking at these matrices!

An isomorphic (i.e., faithful) representation should have as many different matrices as there are group elements, and those matrices should obey the group's multiplication table.
 
Thanks Ben, you seem to be able to describe these things in a way I can understand more easily. The text I reading at the moment is rather formal and I should probably find a better one. Please correct me if I'm wrong here, but would it be right to say that the group of all rotations in three dimensional space is 'isomorphic' to SO(3)?
 
Yes, exactly.
 
The elements of SO(3) must have determinant 1. Why does it need to be so ? Can't it just be isomorphic to O(3) ?
 
I believe it is related to the fact that a rotation matrix with a determinant of -1 represents an 'improper' rotation, and a determinant of 1 represents a 'proper' rotation.

http://en.wikipedia.org/wiki/Improper_rotation" .
 
Last edited by a moderator:
Please forgive me. I'm also trying to understand this group representation.

Is it difficult to prove that the group of all rotations in three dimensional space is 'isomorphic' to SO(3)?

Rotations in 2D surely is isomorphic to SO(2). Rotation about the z-axis can be represented as
[tex] \left(\begin{array}{cc}cos\theta&sin\theta\\-sin\theta&cos\theta\end{array}\right)[/tex]
which is isomorphic to SO(2).
 

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