Past exam of the groups and representations course of the University of Oxford / 2024 / 3 / e

For each complex coordinate, exchanging the factors of a wedge product of differential forms reverses its sign, givingConsequently the Kähler form is $\omega =d{x}_{14}+d{x}_{25}+d{x}_{36}$, and $\omega \wedge d{x}_7=d{x}_{147}+d{x}_{257}+d{x}_{367}$. Expanding the holomorphic volume form givesThe sign of $d{x}_{426}$ changes on exchanging $4,2$, while reordering $4,5,3$ to $3,4,5$ requires two exchanges. This verifies every sign inUnder $z\mapsto Uz$, unitarity preserves $\omega$:The determinant is the factor by which a transformation acts on a top exterior power, so ${\Omega }^{\prime }=(\det U)\Omega =\Omega$ for $U\in SU(3)$. Extend the real embedding by fixing $y$. It preserves $\omega$, $dy$ and $\Omega$, and therefore $\varphi$. ThusThe induced seven-dimensional representation is exactly the restriction already computed in c. In the conventional complex notation its branching rule isOver 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.