Real representation

A real representation of a group $G$ is a homomorphism $G\to \mathrm{GL}(V)$ for a real vector space $V$.

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.

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 $n$.

For $C_n=⟨g⟩$, the irreducible real representations are the trivial line, the sign line $g\mapsto -1$ when $n$ is even, and the pairwise nonisomorphic real planes on which $g$ rotates through $2\pi k/n$ for $1\le k<n/2$.

The real regular representation of $C_n$ is the direct sum of the irreducible real representations of a finite cyclic group, each once. A real Fourier basis consists of the constant vector, the alternating vector when $n$ is even, and cosine-sine pairs for frequencies $1\le k<n/2$.