Understanding the 4 dimensional irrep of $ SU_2 $ The 2d irrep of $ SU_2 $ is just the defining irrep
$$
SU_2:=\left\{ 
\begin{bmatrix}
\alpha & \beta \\ 
-\overline{\beta} & \overline{\alpha}
\end{bmatrix} : |\alpha|^2+|\beta|^2=1
\right\}
$$
The 3d irrep of $ SU_2 $ is the subgroup
$$
SO_3(\mathbb{R})
$$
of $ U_3 $. This is the adjoint representation of $ SU_2 $.
I want to better understand the $ 4d $ irrep of $ SU_2 $. Is there some well known subgroup of $ U_4 $ that is the image of a $ 4d $ irrep of $ SU_2 $? For example can we think of the $ 4d $ irrep of $ SU_2 $ as a subgroup of $ SU_2 \otimes SU_2 $? Or perhaps as a subgroup of the $ 10 $ dimensional symplectic subgroup $ Sp_2 $ of $ U_4 $?
 A: In general, the $(n+1)$-dimensional irrep of $SU(2)$ is the $n^{th}$ symmetric power $S^n(\mathbb{C}^2)$, which can be described explicitly as the space of homogeneous polynomials of degree $n$ in two variables. So the $4$-dimensional irrep is $S^3(\mathbb{C}^2)$, with basis $\{ x^3, x^2 y, xy^2, y^3 \}$, where $\left[ \begin{array}{cc} \alpha & \beta \\ - \overline{\beta} & \overline{\alpha} \end{array} \right]$ acts via extending the action $x \mapsto \alpha x - \overline{\beta} y$ and $y \mapsto \beta x + \overline{\alpha} y$ by multiplication. (Here I'm using the convention that $x$ and $y$ correspond to the standard basis of $\mathbb{C}^2$. There's another convention where we consider polynomial functions on $\mathbb{C}^2$ rather than the symmetric power, which is the dual of this representation, but all these reps are self-dual anyway.)
This representation is quaternionic; to see this we use the fact that the defining representation $\mathbb{C}^2$ is quaternionic, so admits an antilinear map $J : \mathbb{C}^2 \to \mathbb{C}^2$ satisfying $J^2 = -1$. Then $J^{\otimes 3} : (\mathbb{C}^2)^{\otimes 3} \to (\mathbb{C}^2)^{\otimes 3}$ is again an antilinear map squaring to $-1$ and it induces a map with that same property on $S^3(\mathbb{C}^2)$. So the corresponding image of $SU(2)$ in $SU(4)$ is contained in a conjugate of $Sp(2)$ but not of $SO(4)$. Presumably the corresponding map $Sp(1) \to Sp(2)$ has some nice quaternionic description but I don't know it.
