Question about the affine general linear group.

May 2010
4
0
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.
 
May 2010
95
38
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).