show that the symmetric real matrix $R$ is in fact a projection operator; that is, it has only two distinct eigenvalues, 0 and 1. According to Eq. (\[rconditionshort\]), a set $\vec{r}$ of transition rates represents a Hamiltonian evolution if and only if it lies in the eigenvalue-1 subspace of $R$. Moreover, if we start with any real $N^4$-component vector $\vec{v}$ and apply $R$ to $\vec{v}$, the result will be a legitimate set of transition rates associated with some Hamiltonian evolution. (Possibly the result will be the zero vector, but this vector does indeed define a legitimate set of transition rates.)

For odd prime values of $N$, the three-point structure function $\Gamma_{\alpha\beta\gamma}$ takes a particularly simple form, and we can use this fact to write down the condition (\[rcondition\]) more explicitly. Specifically, we have $$\label{oddGamma}
\Gamma_{\alpha\beta\gamma} = \frac{1}{N} \exp \left[ -\frac{4 \pi i}{N} \big(\langle\alpha,\beta\rangle + \langle\beta,\gamma\rangle + \langle\gamma, \alpha\rangle\big) \right],$$ where again $\langle\alpha, \beta\rangle = \alpha_2 \beta_1 - \alpha_1 \beta_2$. Plugging this expression into Eq. (\[rcondition\]) and doing the sum over $\beta$, we get $$\label{rcondition4}
r_{\alpha\gamma} = \frac{1}{N^2} \sum_\zeta (r_{\alpha+\zeta, \gamma + \zeta} - r_{\gamma + \zeta, \alpha+\zeta})
\cos \left[ \frac{4 \pi}{N} \langle\alpha - \gamma, \zeta\rangle \right] .$$ If we now 