= Solution
For each complex coordinate, exchanging the factors of a <wedge product of differential forms> reverses its sign, giving
$$
dz_k\wedge d\bar z_k
=(dx_k+i\,dx_{k+3})\wedge(dx_k-i\,dx_{k+3})
=-2i\,dx_k\wedge dx_{k+3}.
$$
Consequently the <Kähler form> is $\omega=dx_{14}+dx_{25}+dx_{36}$, and $\omega\wedge dx_7=dx_{147}+dx_{257}+dx_{367}$. Expanding the <holomorphic volume form> gives
$$
\begin{aligned}
\operatorname{Re}\bigl((dx_1+i\,dx_4)\wedge(dx_2+i\,dx_5)\wedge(dx_3+i\,dx_6)\bigr)
&=dx_{123}-dx_{156}-dx_{426}-dx_{453}\\
&=dx_{123}-dx_{156}+dx_{246}-dx_{345}.
\end{aligned}
$$
The sign of $dx_{426}$ changes on exchanging $4,2$, while reordering $4,5,3$ to $3,4,5$ requires two exchanges. This verifies every sign in
$$
\boxed{\phi=\omega\wedge dy+\operatorname{Re}\Omega,\qquad y=x_7.}
$$
Under $z\mapsto Uz$, <unitarity> preserves $\omega$:
$$
\omega' =\frac i2\sum_{a,b}\left(\sum_kU_{ka}\bar U_{kb}\right)dz_a\wedge d\bar z_b
=\omega.
$$
The <determinant> is the factor by which a transformation acts on a top <exterior power>, so $\Omega'=(\det U)\Omega=\Omega$ for $U\in SU(3)$. Extend the real embedding by fixing $y$. It preserves $\omega$, $dy$ and $\Omega$, and therefore $\phi$. Thus
$$
\boxed{U\longmapsto\operatorname{diag}(\rho(U),1)
\quad\text{embeds }SU(3)\text{ into }G_2.}
$$
The induced seven-dimensional <representation> is exactly the restriction already computed in <3/c>. In the conventional complex notation its <branching rule> is
$$
\boxed{\mathbf7_{G_2}\downarrow SU(3)=\mathbf3\oplus\bar{\mathbf3}\oplus\mathbf1.}
$$
Over the <real numbers>, the six-dimensional summand is the underlying real space of the complex <fundamental representation>, and the last real line is the fixed $y$ axis.
Back to article page
