Hi,

I'm trying to investigate the complex representations of AGL(2,3) and ASL(2,3), respectively the affine general linear group and the affine special linear group of order 2 over the field of 3 elements, but I'm slightly at a loss as to how to go about it - specifically, I am trying to work out the minimal degree of a faithful complex representation of AGL(2,3), and how many non-isomorphic representations there are of this degree, and then do the same thing for ASL(2,3).

Does anyone have any hints as to how to make a start on this? I've done a course on Representation Theory, but I've never really seen a question like this. Any help would be greatly appreciated.

Thanks,

Jonathan.

Hey, this is my attempt:

Let \(\displaystyle \text{AGL}(2, K)\), where \(\displaystyle K=\mathbb{F}_3\), be an affine general linear group of degree 2 over the field of order 3. It is the external semidirect product of the vector space \(\displaystyle K^2\) by \(\displaystyle \text{GL}(2, K)\) by linear transformations.

Since the action of \(\displaystyle \text{GL}(2, K)\) is natural on \(\displaystyle K^2\), we have \(\displaystyle \text{AGL}(2, K) = K^2 \rtimes \text{GL}(2, K)\), where \(\displaystyle K=\mathbb{F}_3\). See

here.

The matrix representation of \(\displaystyle \text{AGL}(2, K)\) is as follows:

\(\displaystyle \left(\begin{array}{c|c} M & v \\ \hline 0 &1 \end{array} \right)\),

where M is an \(\displaystyle 2 \times 2\) matrix over K and v is an \(\displaystyle 2\times 1\) column vector.

I think the number of non-isomorphic faithful matrix representations of \(\displaystyle \text{AGL}(2, K)\) and the corresponding special affine general linear group are the orders of \(\displaystyle GL(2, K)\) and \(\displaystyle SL(2,K)\), where \(\displaystyle K=\mathbb{F}_3\) (see

here to find the order of general linear group over a finite field).