= Realification of a complex representation
{title2=$\mathcal R(X+iY)=\begin{pmatrix}X&-Y\\Y&X\end{pmatrix},\quad\chi_\mathbb R=2\operatorname{Re}\chi_\mathbb C$}
A complex <group representation> $\rho:G\to\operatorname{GL}(n,\mathbb C)$ 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> $\chi_\mathbb R(g)=2\operatorname{Re}\chi_\mathbb C(g)$. 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.
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