Permutation Representations on Cohomology of Toric Varieties

Abstract

Let $G$ be a finite group acting properly by lattice automorphisms on a complete simplicial fan $\Sigma$. An open question due to Stanley asked whether the (ungraded) representation carried by the cohomology $H^{\ast }(X_{\Sigma })$ of the associated toric variety $X_{\Sigma }$ 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.

arXiv comment15 pages, comments are welcome!