Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 10C / iv / Solution

The map $x\mapsto -x$ is an involution, soIf $\lambda$ is an eigenvalue of ${\mathrm{QFT}}_N$, then ${\lambda }^4=1$. Hence every eigenvalue belongs toBecause $X^4-1$ has distinct roots, the transform is diagonalizable; which fourth roots occur and their multiplicities depend on $N$.