Projective representaions of $(\mathbb{Z}/3\mathbb{Z})^2$

Question

I have a very short question: is there a faithful projective representaion $\rho: \mathbb{Z}/3\mathbb{Z}\times \mathbb{Z}/3\mathbb{Z}\to {\rm PGL}(4,\mathbb R)$?

Thanks!

Answer 1

Following Derek's hint, we first pick a faithful representation $\mathbb{Z}/3\mathbb{Z} \rightarrow GL(2, \mathbb{R})$.

Send $1$ to

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

You can check that this is an element of order 3, so it gives us a representation. It's also obviously faithful.

Then, define a representation of $\mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$ by sending the first generator to the above matrix in the first $2 \times 2$ block and the second generator to the above matrix in the second $2 \times 2$ block. This gives us a faithful representation of the group in which no element is mapped to a scalar matrix so it remains faithful after taking quotients.