How to Identify a 1-Dimensional ℂG-Submodule in V≅ℂ4?

[tomy123]

Let ψ(a) and ψ(b) be the matrix representation of the generators a,b of a group G.
How can we find 1-dimensional ℂG-submodule of the ℂG-module V≅ℂ⁴ ?

 

[jvicens]

To find a 1-dimensional ℂG-submodule of the ℂG-module V≅ℂ4, we can use the following steps:

1. First, we need to determine the action of the generators a and b on the ℂ4 vector space. This can be done by expressing ψ(a) and ψ(b) as matrices and multiplying them with a vector from ℂ4.

2. Next, we can use the fact that a 1-dimensional submodule is spanned by a single vector. So, we need to find a vector in ℂ4 that is invariant under the action of both a and b.

3. One way to do this is by finding the eigenvectors of ψ(a) and ψ(b). These are vectors that remain unchanged (up to a scalar multiple) when acted upon by the respective matrices.

4. Once we have found the eigenvectors, we can check if they are invariant under the action of both a and b. If they are, then we have found our 1-dimensional submodule.

5. If we are unable to find a suitable eigenvector, we can also try to construct a vector that is a linear combination of the eigenvectors. This vector may also be invariant under the action of a and b, and thus form a 1-dimensional submodule.

Overall, finding a 1-dimensional ℂG-submodule of the ℂG-module V≅ℂ4 involves understanding the action of the generators on the vector space and finding vectors that are invariant under this action.