= Exterior and symmetric squares of the seven-dimensional G2 representation
For $V=L(\omega_1)$ with the <G2 root system> numbered short-first,
$$
\Lambda^2V=L(\omega_2)\oplus L(\omega_1),\qquad S^2V=L(2\omega_1)\oplus L(0).
$$
In the chain basis of the <crystal of the seven-dimensional G2 representation>, normalize $v_1=f_1v_0$, $v_2=f_2v_1$, $v_3=f_1v_2$. Then $v_0\wedge v_1$ and $v_0\wedge v_3-2v_1\wedge v_2$ are nonzero <highest-weight vectors> of weights $\omega_2$ and $\omega_1$; the identities $e_1v_1=v_0$, $e_1v_3=2v_2$, $e_2v_2=v_1$ verify both raising conditions. The <Weyl complete reducibility theorem> and <G2 dimension polynomial> exhaust the twenty-one dimensions of the <exterior square>. In the <symmetric square>, $v_0^2$ generates the twenty-seven-dimensional summand. Self-duality supplies a <Lie-invariant bilinear form>, which is symmetric because a <nondegenerate> <alternating bilinear form> cannot have odd <dimension>; its inverse gives the remaining invariant line.
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