Understanding Group Representations in Group Theory

In summary, a representation of a group is a set of square matrices that follows the group's multiplication table. A faithful representation is an isomorphism where the matrices behave the same way as the group. An isomorphic representation should have as many matrices as there are group elements and they should obey the group's multiplication table. The group of all rotations in three dimensional space is isomorphic to SO(3) because it represents proper rotations. It is related to the fact that a rotation matrix with a determinant of -1 represents an improper rotation, while a determinant of 1 represents a proper rotation. It is also possible to represent rotations in 2D using SO(2), as shown with the rotation about the z-axis. Proving the
  • #1
jimmy.neutron
17
0
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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  • #2
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.
 
  • #3
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?
 
  • #4
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.
 
  • #5
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)?
 
  • #6
Yes, exactly.
 
  • #7
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) ?
 
  • #8
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:
  • #9
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).
 

What are group representations?

Group representations are mathematical objects used to describe the symmetries of a group. They are used to study the behavior of a group under transformations and can help us understand the structure and properties of the group.

Why are group representations important?

Group representations have many applications in mathematics, physics, and other sciences. They can be used to study symmetry in geometric objects, analyze the behavior of particles in quantum mechanics, and understand the properties of molecules and crystals.

What are the different types of group representations?

The most common types of group representations are matrix representations, where elements of the group are represented by matrices, and permutation representations, where elements are represented by permutations of a set. Other types include character representations and projective representations.

How are group representations related to group theory?

Group representations are an important tool in group theory, as they can help us understand the structure and behavior of a group. In fact, group representations are often used to define and classify different types of groups, such as finite groups, Lie groups, and topological groups.

How do you construct a group representation?

To construct a group representation, we first need to choose a vector space on which the group will act. Then, we define a homomorphism from the group to the group of invertible linear transformations on the vector space. This homomorphism will be the group representation, and it can be represented by matrices or other mathematical objects.

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