Sym$^2\mathbb{C}^2$ as the unique 3-dimensional irrep of $\operatorname{SU}(2)$

Question

In this script (Link) regarding GUTs it is stated that the unique 3-dimensional complex representation of $\operatorname{SU}(2)$ up to isomorphism is given by Sym$^2\mathbb{C}^2$, the symmetric tensors of rank 2. I was wondering however whether this is indeed true. The author asserts that this representation can also be understood as the Adjoint representation on $\mathbb{C} \otimes \mathfrak{su}(2) \simeq \mathfrak{sl}(2,\mathbb{C})$. The adjoint representation of $\mathfrak{sl}(2,\mathbb{C})$ is however the special linear tensors of rank 2 and not just the symmetric tensors of rank 2. Am I missing something or is the statement wrong?

Answer 1

If $$V_L~\cong~\mathbb{C}^2\tag{1}$$ denotes the fundamental/defining/left Weyl/spin-$(\frac{1}{2},0)$ representation of the Lie group $\operatorname{SL}(2,\mathbb{C})$, then $$\begin{align} \{M\in{\rm Mat}_{2\times 2}(\mathbb{C}) \mid M^t=M\} ~\cong~& {\rm Sym}^2V_L\cr ~\cong~& \mathfrak{sl}(2,\mathbb{C})\cr ~\cong~&\{M\in{\rm Mat}_{2\times 2}(\mathbb{C}) \mid {\rm tr}M=0\}\cr ~\cong~&\mathfrak{su}(2)\otimes \mathbb{C} \end{align}\tag{2}$$ is the adjoint/spin-$(1,0)$ representation, cf. e.g this Phys.SE post.