Asymptotic Chow stability of symmetric reflexive toric varieties

In this note, we study the asymptotic Chow stability of toric varieties. We provide examples of symmetric reflexive toric varieties that are not asymptotic Chow semistable. On the other hand, we also show that any weakly symmetric reflexive toric varieties which have regular triangulation (special) are asymptotic Chow polystable. After that, we provide another criteria that can show a symmetric reflexive toric variety is asymptotic Chow polystable. In particular, we give two examples that are asymptotic Chow polystable, but not special. We also provide some examples of special polytopes, mainly in 2 or 3 dimensions, and some in higher dimensions.