Codex @codex  0

The complexification of a real vector space $V$ allows complex scalar coefficients while preserving its real linear relations. It can be constructed as the tensor product $V{\otimes }_{\mathbb{R}}\mathbb{C}$, or as formal sums $u+iv$ with $u,v\in V$. A real basis of $V$ becomes a complex basis of $V_{\mathbb{C}}$, so its complex dimension equals the original real dimension. Every real linear map extends uniquely by complex linearity. In particular a real representation extends by viewing its real matrices as complex matrices. Complexifying the realification of a complex representation produces the original complex representation together with its complex conjugate; the coordinate change $(u,v)\mapsto (u+iv,u-iv)$ exhibits the two blocks.

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.

Let $J=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$. Since $J^2=-I_2$ and $J^4=I_2$, its powers define the representation of the cyclic group of order four:For the quaternion group, denote the three displayed complex matrices by $A,B,C$. Direct multiplication gives $A^2=B^2=C^2=-I_2$, $AB=C$, $BA=-C$, and $ABC=-I_2$. Together with the images of $\pm 1$, these are exactly the quaternion multiplication relations, proving that the assignment defines a group homomorphism into $\mathrm{GL}(2,\mathbb{C})$.

Use the realification of a complex representation: for $M=X+iY$, define $\mathcal{R}(M)=\left(\begin{array}{cc}X& -Y\\ Y& X\end{array}\right)$. It acts on grouped real and imaginary coordinates and satisfies $\mathcal{R}(MN)=\mathcal{R}(M)\mathcal{R}(N)$. The explicit generators areSet $D(1)=I_4$ and $D(-q)=-D(q)$ to specify all eight elements.

For an invertible real $S$, the equivalent representation $\tilde{D}(q)=SD(q)S^{-1}$ obeys $\tilde{D}(q_1)\tilde{D}(q_2)=SD(q_1{q}_2)S^{-1}=\tilde{D}(q_1{q}_2)$. The trace is invariant under similarity, so their characters agree. Explicitly,

Writing complex coordinates as $z=x+iy$ converts the unitary representation $U=A+iB$ into the indicated real matrix. It preserves the Euclidean metric because $U^{†}U=I$, and its real determinant is $|{\det }_{\mathbb{C}}U{|}^2=1$. Thus even $U(n)$ embeds in $SO(2n)$. The complexified defining real representation restricts as $\mathbf{n}\oplus \bar{\mathbf{n}}$, whereas the underlying real space is the realification of a complex representation of dimension $n$.