In the book Gauge Fields, Knots and Gravity, spin "j" representation is described as homomorphisms from SU(2) to the General linear group of space of polynomials of degree $2j$ in $\mathbb{C}$ in one specific way namely $(\rho(g)f)(v)=f({g}^{-1}(v))$, Here $v=(x,y)$ is an element of $\mathbb{C}^2$, $f$ is a degree $2j$ polynomial and $g$ is an element of $SU(2)$.
On Wikipedia spin group is defined as follows.
The spin group $Spin(n)$ is the double cover of the special orthogonal group $SO(n) = SO(n, R)$, such that there exists a short exact sequence of Lie groups (when $n \neq 2$).
$ {1} \rightarrow Z_2 \rightarrow Spin(n) \rightarrow SO(n) \rightarrow 1 $
Could someone describe how these two definitions are related?
Thanks