 Solution

A group representation is a homomorphism $D:G\to \mathrm{GL}(V)$ into invertible linear maps. It is a faithful representation when it is injective, equivalently when its kernel contains only the identity. A faithful real two-dimensional representation of the cyclic group of order four isThese are rotations of the plane through $0,\pi /2,\pi ,3\pi /2$. Since $J^2=-I$ and $J^4=I$, their multiplication agrees with the group, and all four matrices are distinct.

For any invertible $S$, set $\tilde{D}(g)=SD(g)S^{-1}$. Thenso this also is a group representation. The character of a representation is ${\chi }_D(g)=\mathrm{tr}D(g)$. For the displayed quarter-turn representation, its values on $1,i,-1,-i$ are $2,0,-2,0$.