What are the Matrices of the Regular Representation for the Cyclic Group C2?

[PhysKid24]

Hi,

I have a question regarding group theory. For the cyclic group C2 with elements e and a, what are the matrices of the regular representation? How do you find this? How would I reduce this representation into irreducible representation? Lastly, how do I find a matrix which brings the regualr representation into block diagonal form? Thanks.

 

[matt grime]

At last, some representation theory to do (my area).

The regular rep is two dimesional, saw with basis e and f corresponding to 1 and g in G resp.

1 is sent to the identity, natch, and g sends e to f and f to e, so it is the matrix

$$\left( \begin{array}{cc}0&1\\1&0\end{array} \right)$$

i'm sure i can leave you to diagonalize that, over R or C (but not a field of characteristic 2).

 

[M98Ranger]

Hi there,

The regular representation of a group is the most basic representation, where each element of the group is represented by a matrix. In the case of the cyclic group C2, with elements e and a, the regular representation would have two matrices: one for e and one for a.

To find the matrices for the regular representation, you can use the fact that the regular representation preserves the group operation. This means that the matrix for the identity element e should be the identity matrix, and the matrix for a should be such that when multiplied by itself, it gives the identity matrix. In this case, a simple 2x2 matrix with 1 in the top left and bottom right corners, and 0 in the other entries, would suffice.

To reduce the regular representation into irreducible representations, you can use the character table of the group. Each irreducible representation will have a unique character, which is the trace of its corresponding matrix. By finding the characters of the regular representation and comparing them to the character table, you can determine which irreducible representations are present in the regular representation.

To bring the regular representation into block diagonal form, you would need to find a similarity transformation matrix that diagonalizes the regular representation matrix. This can be done by finding the eigenvalues and eigenvectors of the regular representation matrix, and constructing the similarity transformation matrix from these eigenvectors.

I hope this helps! Let me know if you have any further questions.