Symmetric and K\"ahler--Einstein Fano polygons

We investigate \emph{singular} symmetric and K"ahler–Einstein Fano polytopes. More precisely, we show that every symmetric Fano polytope is K"ahler–Einstein generalizing the work by Batyrev and Selivanova, and study the automorphism groups of symmetric and K"ahler–Einstein Fano polygons in detail. In particular, every finte subgroup of $G{L}_2(\mathbb{Z})$ is an automorphism group of a K"ahler–Einstein Fano polygon.