Understanding Group Representations in Group Theory

[jimmy.neutron]

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

 

[Ben Niehoff]

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.

 

[jimmy.neutron]

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?

 

[Ben Niehoff]

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.

 

[jimmy.neutron]

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)?

 

[Ben Niehoff]

Yes, exactly.

 

[matematikawan]

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) ?

 

[jimmy.neutron]

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" .

 

[matematikawan]

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
<br /> \left(\begin{array}{cc}cos\theta&amp;sin\theta\\-sin\theta&amp;cos\theta\end{array}\right)<br />
which is isomorphic to SO(2).