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In John Terning's book, on pages 14 and 15, there are lists of $\mathcal{N} = 2$ and $\mathcal{N} = 4$ supermultiplets, labeled in terms of the dimensions of the corresponding R-symmetry $d_R$ and spin-symmetry $2j+1$. I want to figure out a way to get all these numbers by Young Tableaus in a systematic way.

Of course, the $\mathcal{N}=2$ case is relatively straightforward. Its clear that the numbers in the labels for $\mathcal{N}=4$ can separately be recovered using

$4_{R} \otimes 4_{R} = 10_R \oplus 6_R$

for $SU(4)_R$ and tensor products of this basic identity with $4_R$. Of course, one can also do the same thing for the $SU(2)$ spin symmetry

$2_{SU(2)} \otimes 2_{SU(2)} = 3_{SU(2)} + 1_{SU(2)}$

But if one writes $(\textbf{R}, 2j+1)$ as the label, then how does one justify

$(\textbf{4}_R, 2)\otimes(\textbf{4}_R, 2) = (\textbf{10}_R,1) \oplus (\textbf{6}_R,3)$

or

$(\textbf{4}_R, 2)\otimes((\textbf{10}_R,1) \oplus (\textbf{6}_R,3)) = (\bar{\textbf{20}}, 2) + (\bar{\textbf{4}},4)$

What I'm asking is: how do you get this particular grouping?

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