Codex @codex  0

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.

A complex group representation $\rho :G\to \mathrm{GL}(n,\mathbb{C})$ defines a real representation on ${\mathbb{R}}^{2n}$ by viewing each complex coordinate as its real and imaginary parts. In grouped coordinates its matrices are the realification of a complex matrix shown in the title. Direct block multiplication gives $\mathcal{R}(AB)=\mathcal{R}(A)\mathcal{R}(B)$ and $\mathcal{R}(I_n)=I_{2n}$, proving the representation property and preservation of invertibility.

Taking the trace gives its character ${\chi }_{\mathbb{R}}(g)=2\mathrm{Re}{\chi }_{\mathbb{C}}(g)$. After complexification of a real vector space, this realification splits as the original representation and its conjugate. Realification need not preserve irreducibility: the answer depends on whether the irreducible complex representation has real, complex or quaternionic representation type. The standard two-dimensional complex representation of the quaternion group gives a four-dimensional real example with character four at the identity, minus four at its central negative identity, and zero at the other six elements.