# What physical meaning do the dimension of Wigner d-matrices have?

Wigner's D-matrices is defined as $D_{m'm}^j(\phi,\theta,\psi)=\langle jm'|R(\phi,\theta,\psi)|jm\rangle$; it produces a square matrix (indices $m$ and $m'$) of dimension $2j+1$. It is also the case that these matrices (for all positive $j$ multiple of $1/2$) are a representation of the rotation group $SU(2)$, which is a double cover of $SO(3)$.

Now, I can see this for the specific case $j=1$ (3x3 matrices representing rotations in 3-dimensional Euclidean space), but what does it mean for other values of $j$? Say for $j=3/2$, does that represent the subgroup of 3-dimensional rotations given some parameter, in a 4-dimensional ambient geometry?

Thanks.

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