EDITED: I learned something from Qiaochu Yuan's answer to my other question on SO(4). I clarify my original question with more information below. My original question is: Does $SO(4)$ have exactly two irreducible 3-dimensional representations that are dual to each other?
The universal cover of SO(4) is ${\mathrm{Spin}}(4)\cong {\mathrm{SU}}(2)\times {\mathrm{SU}}(2)$. An irreducible representation of ${\mathrm{Spin}}(4)$ thus takes the form $V\otimes W$ where $V$ and $W$ are irreducible representations of SU(2). Here I consider representations over the complex numbers. There is exactly one irreducible representation of SU(2) in each dimension. Thus there are exactly two 3-dimensional irreducible representations of ${\mathrm{Spin}}(4)$, namely $V\otimes {\mathbf{1}}$ and ${\mathbf{1}}\otimes V$, where $V$ is the standard representation on SU(2) and ${\mathbf{1}}$ is the trivial representation.
Does the representation $V\otimes {\mathbf{1}}$ descend to $SO(4)$? From the above, we can conclude that either SO(4) has no irreducible 3-dimensional representations or has two of them. Furthermore, if SO(4) has two of them, then these two representations are dual to each other.
Best Answer
Yes, we have that $SO(4)=$Spin$(4)/\mathbb Z_2$ and so we need only worry about whether $-1=-1\otimes1\in $Spin$(4)$ acts trivially, which indeed it does on both the $\mathbf 1$ and $\mathbf 3$.