= Restriction of the seven-dimensional G2 representation to long-root A2
{c}
{title2=$\mathbf7\downarrow_{A_2}=\mathbf3\oplus\overline{\mathbf3}\oplus\mathbf1$}
The seven-dimensional fundamental $G_2$ representation has weights zero and the six short roots. Under its long-root $A_2$ subalgebra, these are the weights of the defining three-dimensional representation, its dual, and the trivial representation:
$$
V_7\downarrow_{A_2}\cong V(\omega_1)\oplus V(\omega_2)\oplus V(0).
$$
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