Unwinding Toric Degenerations and Mirror Symmetry for Grassmannians

The most fundamental example of mirror symmetry compares the Fermat hypersurfaces in P-n and P-n/G, where G is a finite group that acts on P-n and preserves the Fermat hypersurface. We generalize this to hypersurfaces in Grassmannians, where the picture is richer and more complex. There is a finite group G that acts on the Grassmannian Gr(n, r) and preserves an appropriate Calabi-Yau hypersurface. We establish how mirror symmetry, toric degenerations, blow-ups and variation of GIT relate the Calabi-Yau hypersurfaces inside Gr(n, r) and Gr(n, r)/G. This allows us to describe a compactification of the Eguchi-Hori-Xiong mirror to the Grassmannian, inside a blow-up of the quotient of the Grassmannian by G.