Q: $(1,1)$ representation of $SL(2,\mathbb{C})$ How do you prove that the $(1,1)$ representation of the $SL(2,\mathbb{C})$ group acts on symmetric, traceless tensors of rank 2?

A: Upon a better look, it was not solved explicitely in a textbook. Start first with the Clebsch-Gordan decomposition of a tensor product of $SL(2,\mathbb{C})$ in a formal manner as:
$$\left(\frac{1}{2},\frac{1}{2}\right) \otimes \left(\frac{1}{2},\frac{1}{2}\right) =  \left[\left(0,0\right)\oplus \left(1,1\right)\right]_S \oplus \left[\left(1,0\right)\oplus \left(0,1\right)\right]_A $$ 
In the LHS you have $\psi_{a\dot{a}}\otimes \phi_{b\dot{b}} \equiv \chi_{a\dot{a}b\dot{b}}$, to which you can use the Infeld-van der Waerden symbol $\sigma^{\alpha}_{~a\dot{a}}$ and $ \left(\sigma_{\alpha\beta}\right) _{a} ^{~b}$ to go to Lorentz indices which could be manipulated in the RHS to the sum (in a vector space sense) of a trace of 2nd rank tensor, then the traceless symmetric part of that tensor and the fully antisymmetrical part of that tensor.  The full details with calculations in case of Maxwell and Einstein gravitational fields are found in chapters 5 and 6 of Moshe Carmeli and Shimon Malin's Theory of Spinors: An Introduction, WS, 2000.
