Representations of cyclic groups acting on complete simplicial fans

Let $sigma$ be a complete simplicial fan in finite dimensional real Euclidean space $V$, and let $G$ be a cyclic subgroup of $GL(V)$ which acts properly on $\sigma$. We show that the representation of $G$ carried by the cohomology of $X_{sigma}$, the toric variety associated to $sigma$, is a permutation representation.