Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 19I / b

Let $G=C_3=⟨g⟩$ and let $e_0,e_1,e_2$ be the standard basis of its real regular representation, with $g{e}_j=e_{j+1}$ modulo three. SetBoth are invariant, they are orthogonal, andThe line $U$ is the trivial representation. On the plane $W$, the generator acts as a rotation through $2\pi /3$. Such a rotation has no real eigenline, so $W$ has no nonzero proper invariant real subspace and is irreducible.

The two irreducible real modules $U$ and $W$ are nonisomorphic, so the kernel-image argument from part (a), which is valid over any field, givesEvery endomorphism of the one-dimensional trivial module is scalar:

Identify the oriented plane $W$ with $\mathbb{C}$, so that $g$ acts by multiplication by $e^{2\pi i/3}$. A real-linear map commutes with this rotation exactly when it is multiplication by a complex number. Equivalently,where $J$ is rotation through $\pi /2$. Taking $J$ gives a nonscalar intertwining endomorphism of an irreducible real representation. This is the Failure of the scalar conclusion of Schur lemma over the real numbers.