I've been working through some computations involving representations of the Lorentz group (now using the fantastic Ticciati QFT textbook).
After some work, Ticciati gives the following formula $$D^{j_{1},j_{2}}(X_{i})=D^{j_{1}}(T_{i})\otimes \mathbf{I}_{2j_{2}+1}+\mathbf{I}_{2j_{1}+1}\otimes D^{j_{2}}(T_{i}),$$ where $X_{i}$ are the generators of the Lorentz group written in the mathematicians convention without the extra factor of i, and the $T_{i}$ are the $\mathfrak{su}(2)$ matrices.
I've computed $D^{0,1/2}(X_{k})$ and $D^{1/2,0}(X_{k})$, obtaining $-\frac{i}{2}\sigma_{k}$ in each case. This result agrees with what Ticciati gets in eq (6.7.9).
The issue arrives when I compute $D^{1/2,1/2}$. I know that I will have to perform a change of basis using the Clebsch-Gordon coefficients, however, I only end up with the correct matrix if I add a complex conjugate to the formula Ticciati gives: $$D^{j_{1},j_{2}}(X_{i})=D^{j_{1}}(T_{i})\otimes \mathbf{I}_{2j_{2}+1}+\mathbf{I}_{2j_{1}+1}\otimes (D^{j_{2}}(T_{i}))^{*}.$$
I found another post: Proof that $(1/2,1/2)$ Lorentz group representation is a 4-vector Which does the same, however, the author of this post has not explained why this complex conjugate appears.
I tried to derive the formula using the complexified Lorentz algebra $$A_{k}=\frac{1}{2}(X_{k}+iB_{k}), \qquad C_{k}=\frac{1}{2}(X_{k}-iB_{k}),$$ and then embedding reps into the product space $\mathbf{C}^{(2j_{1}+1)(2j_{2}+1)}$ by writing $$D^{j_{1},j_{2}}(A_{k})=D^{j_{1}}(T_{k})\otimes \mathbf{I}_{2j_{2}+1}$$ and $$D^{j_{1},j_{2}}(C_{k})=\mathbf{I}_{2j_{1}+1}\otimes D^{j_{2}}(T_{k}).$$ Unfortunately I still get the same issue! There must be something that I'm not understanding! Any help would be appreciated.
*Edit: I give here an explicit calculation for $X_{1}$ using the complex conjugate.
$D^{1/2,1/2}(X_{1})=\frac{-i}{2}\sigma_{1}\otimes \mathbf{I}_{2}+\mathbf{I}_{2}\otimes (\frac{-i}{2}\sigma_{1})^{*}=\begin{pmatrix} 0 & 0 & \frac{-i}{2} & 0 \\ 0 & 0 & 0 & \frac{-i}{2} \\ \frac{-i}{2} & 0 & 0 & 0 \\ 0 & \frac{-i}{2} & 0 & 0 \\ \end{pmatrix}+\begin{pmatrix} 0 & \frac{i}{2} & 0 & 0 \\ \frac{i}{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{i}{2} \\ 0 & 0 & \frac{i}{2} & 0 \\ \end{pmatrix}$
Then the author in the post I linked above uses the following matrix to change basis (if someone could explain where this matrix is obtained that would be very helpful!): $U=\frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ 0 & i & -i & 0 \\ 1 & 0 & 0 & -1 \\ \end{pmatrix}$
Then I get $U^{-1}D^{1/2,1/2}(X_{1})U=\begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \\ \end{pmatrix}=-iJ_{1}$
