$\DeclareMathOperator\SO{SO}\DeclareMathOperator\Spin{Spin}$What is the exact relationship between the finite dimensional representations of the group $\SO(n)$ and its covering group $\Spin(n)$? More precisely, what are examples of representations of $\Spin(n)$ that do not factor through a representation of $\SO(n)$? Can we give a complete description of such representations?



Relationship between the representation theory of $\operatorname{Spin}(n)$ and $\operatorname{SO}(n)$
$\def\Spin{\text{Spin}}$The cover $\Spin(n) \to SO(n)$ is $2$ to $1$, the nontrivial element kernel is a central element of $\Spin(n)$ which I'll call $z$. Since $z$ is central, it acts on any irrep of $\Spin(n)$ by $\pm 1$; if $z$ acts by $1$, then the irrep factors through $SO(n)$, if $z$ acts by $-1$, then it does not.


Irreps of Lie groups are usually described in terms of high weight vectors, which are characters of maximal tori. For notational simplicity, I'll take $n = 2m$ to be even. Then a maximal torus $T$ of $SO(2m)$ looks like
$$\begin{bmatrix} 
\cos \theta\_1 & - \sin \theta\_1 &&&&& \\
\sin \theta\_1 & \cos \theta\_1 &&& \\
&& \cos \theta\_2 & - \sin \theta\_2 &&& \\
&& \sin \theta\_2 & \cos \theta\_2 &&& \\
&& && \ddots && \\
&&&&& \cos \theta\_m & \sin \theta\_m \\ 
&&&&& - \sin \theta\_m & \cos \theta\_m \\
\end{bmatrix}$$
so a character of $T$ looks like $e^{i (k\_1 \theta\_i+ \cdots + k\_m \theta\_m)}$ for some $(k\_1, \ldots, k\_m)$ in $\mathbb{Z}^m$. Such a character is dominant if $k\_1 \geq k\_2 \geq \cdots \geq k\_{m-1} \geq |k\_m|$. So irreps are indexed by $m$-tuples of integers $(k\_1, \ldots, k\_m)$ with $k\_1 \geq k\_2 \geq \cdots \geq k\_{m-1} \geq |k\_m|$.


Letting $\widehat{T}$ be a maximal torus for $\Spin(n)$, we have a short exact sequence
$$1 \to \langle z \rangle \longrightarrow \widehat{T} \longrightarrow T \to 1.$$
Characters of $\widehat{T}$ can be thought of as $m$-tuples $(k\_1, \ldots, k\_m)$ which lie in $\mathbb{Z}(1/2, 1/2, \ldots, 1/2) + \mathbb{Z}^m$, with the factors that have integer entries factoring through $T$ and the ones with half-integer entries not factoring. Again, a character is dominant if $k\_1 \geq k\_2 \geq \cdots \geq k\_{m-1} \geq |k\_m|$.


The simplest example of a representation of $\Spin(n)$ which doesn't factor through $SO(n)$ is the irrep with high weight $(1/2, 1/2, \ldots, 1/2)$; call it $W$. There isn't a simple description of this representation, but one thing that I find helpful is that there is a nonzero map $W \otimes W \longrightarrow \bigwedge^{m} \mathbb{R}^{2m}$. If you compute the matrices for $SO(2m)$ acting on $\bigwedge^{m} \mathbb{R}^{2m}$, you'll see that lots of the matrix entries are determinants of $m \times m$ skew symmetric matrices; the matrix entries for $W$ are the square roots of these determinants, meaning certain Pfaffians of $m \times m$ skew symmetric matrices.


There is no simple construction of $\Spin(n)$, but the usual route is by the Clifford algebra and can be found in books like Fulton and Harris's representation theory.


