Let a finite group act on a smooth projective toric variety via lattice automorphism. We show that its cohomology is a permutation representation: it is induced from a group action on a set. This answers a question of Stanley, for all smooth projective toric varieties, without the usual properness assumption. The proof uses ideas from mirror symmetry. Joint with Tao Gui, Chushi Qin, and Kaizhe Shen.