Asymptotic Chow stability of symmetric reflexive toric varieties

Abstract In this note, we study the asymptotic Chow stability of symmetric reflexive toric varieties. We provide examples of symmetric reflexive toric varieties that are not asymptotically Chow semistable. On the other hand, we also show that any weakly symmetric reflexive toric varieties which have a regular triangulation (so are special) are asymptotically Chow polystable. Furthermore, we give sufficient criteria to determine when a toric variety is asymptotically Chow polystable. In particular, two examples of toric varieties are given that are asymptotically Chow polystable, but not special. We also provide some examples of special polytopes, mainly in two or three dimensions, and some in higher dimensions.