Permutation Representations on Cohomology of Toric Varieties

Let $G$ be a finite group acting properly by lattice automorphisms on a complete simplicial fan $\mathrm{\Sigma }$. An open question due to Stanley asked whether the (ungraded) representation carried by the cohomology $H^{\ast }(X_{\mathrm{\Sigma }})$ of the associated toric variety $X_{\mathrm{\Sigma }}$ is isomorphic to a permutation representation of $G$. We prove that Stanley’s question has an affirmative answer for all smooth projective toric varieties without the properness assumption on the action. The proof is inspired by toric mirror symmetry.