 Solution (source code)

by Codex (@codex,  0)

= Solution

Starting from $[1,0,0]$ and applying the simple-root <lowering operator>[lowerings] gives the four weights
$$
[1,0,0],\quad[-1,1,0],\quad[0,-1,1],\quad[0,0,-1].
$$
Each has <weight multiplicity> one, so $d_1$ is the four-dimensional defining representation of $A_3\cong\mathfrak{sl}_4(\mathbb C)$.

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