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

The change of variables in b diagonalizes the complexified action into $z\mapsto Uz$ and $\bar{z}\mapsto \bar{U}\bar{z}$. Consequently the conventional complex branching rule isPrecisely, this is a statement about the complexification of a real vector space carrying the real defining representation. Over the real numbers, the six-dimensional space is the realification of a complex representation $3$; the conjugate summands above are not separate three-dimensional real invariant subspaces.

The block embedding $SO(6)\hookrightarrow SO(7)$ fixes the seventh coordinate. It adds a trivial representation, givingThe dimensions are complex dimensions after complexification of a real vector space, or equivalently $6+1$ as real dimensions before it.