Representations on the cohomology of hypersurfaces and mirror symmetry

We study the representation of a finite group acting on the cohomology of a non-degenerate, invariant hypersurface of a projective toric variety. We deduce an explicit description of the representation when the toric variety has at worst quotient singularities. As an application, we conjecture a representation-theoretic version of Batyrev and Borisov's mirror symmetry between pairs of Calabi-Yau hypersurfaces, and prove it when the hypersurfaces are both smooth or have dimension at most 3. An interesting consequence is the existence of pairs of Calabi-Yau orbifolds whose Hodge diamonds are mirror, with respect to the usual Hodge structure on singular cohomology.Comment: 34 page

Equivariant Ehrhart theory

Motivated by representation theory and geometry, we introduce and develop an equivariant generalization of Ehrhart theory, the study of lattice points in dilations of lattice polytopes. We prove representation-theoretic analogues of numerous classical results, and give applications to the Ehrhart theory of rational polytopes and centrally symmetric polytopes. We also recover a character formula of Procesi, Dolgachev, Lunts and Stembridge for the action of a Weyl group on the cohomology of a toric variety associated to a root system.Comment: 40 pages. Final version. To appear in Adv. Mat