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(I post this in physics because its about an excercise in the Thinkman book of theory group for quantum physics).

The Group of symmetries of the square (D4) has an order of 8. There are 2 classes in the group (correct me if Im wrong). These classes are: One class made of rotations in the plane of the square, of $0$ (identity), $\pi/2$, $\pi$ and $3 \pi /2$. Also 4 rotations with axis inside the plane: two across the diagonals of the square and two perpendicular to the sides by the middle.

Using the "dimensionality theorem", which says that the order is equal to the sum of the square of the dimensionalities of the possible representations, I get $2^2+2^2=8$, meaning that there are 2 representations of order 2.

One of this representations has the first class represented with these matrices:

$\begin{bmatrix} 1 & 0 \\0 & 1 \end{bmatrix}$, $\begin{bmatrix} 0 & -1 \\1 & 0 \end{bmatrix}$, $\begin{bmatrix} -1 & 0 \\0 & -1 \end{bmatrix}$, $\begin{bmatrix} 0 & -1 \\1 & 0 \end{bmatrix}$

And the second class represented as:

$\begin{bmatrix} -1 & 0 \\0 & 1 \end{bmatrix}$, $\begin{bmatrix} 1 & 0 \\0 & -1 \end{bmatrix}$, $\begin{bmatrix} 0 & 1 \\1 & 0 \end{bmatrix}$, $\begin{bmatrix} 0 & -1 \\-1 & 0 \end{bmatrix}$

But my problem is that the traces of a same class should be the same (correct me again if im wrong), and the first class has 1 matrix of $Tr=2$, 1 of $Tr=-2$ and two of $Tr=0$. The second class has every matrix with $Tr=0$.

I immediately thought that the classes I choose were wrong, and that I have 3 classes. But it doesn't have sense for me because, if I have 3 classes I cant fulfill the dimensionality theorem, as there isn't any sum of square integers which is equal to 8.

What I am missing? There are 2 or 3 classes? Im choosing wrong the members of the classes?

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  • $\begingroup$ For finite groups there is always an identity rep where all the matrices are just 1. So you can be sure that when you selected two 2x2 reps you were not correct. Are there any other ways to make 8 out of sums of squares? Do any of them include 1 as one of the reps? $\endgroup$
    – user93146
    Commented Jun 27, 2018 at 21:55
  • $\begingroup$ By the way, wow! I last studied this stuff in 1982. $\endgroup$
    – user93146
    Commented Jun 27, 2018 at 21:56

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