Permutation Representations on Cohomology of Toric Varieties

Let $G$ be a finite group acting properly by lattice automorphisms on a complete simplicial fan $Σ$. An open question due to Stanley asked whether the (ungraded) representation carried by the cohomology $H^*(X_Σ)$ of the associated toric variety $X_Σ$ is isomorphic to a permutation representation of