Relation between defining representation of SU(4) and spin-3/2 representation of SU(2)

Question

Both the defining representation of $SU(4)$ and the spin-3/2 representation of $SU(2)$ are on $\mathbb C^4$ and unitary. I am wondering if there is any more explicit relations between the two Lie group (or their corresponding Lie algebra). For example, is there some sort of nontrivial Lie algebra homomorphism between the generators of the two Lie algebra?

Answer 1

More generally, the spin $j$ representation of $SU(2)$ with dimension $2j+1$, is just the $2j$-th symmetric power ${\rm Sym}^{2j}(\mathbb{C}^2)$ of the defining representation on $\mathbb{C}^2$. In more concrete terms this is just the action by linear change of variables on binary forms or homogeneous polynomials in two variables, of degree $2j$. This construction gives a homomorphism from $SU(2)$ to $SU(2j+1)$ which you can differentiate at the origin, if you want a Lie algebra version.

For more details, see Section 7 of my article with Chipalkatti "The higher transvectants are redundant", Annales de l'Institut Fourier, Tome 59 (2009) no. 5, pp. 1671-1713.