Symmetry notions for toric Fanos

We survey various notions of symmetry for toric varieties. These notions range from algebraic geometric, complex geometric, representation theoretic, combinatorial, convex geometric, to geometric stability. The main theorem gives the relationship between these notions. While mostly folklore knowledge, this does not seem to be readily available in the literature. Finally, we take the opportunity to give an accessible and simplified proof of Demazure’s 1970 structure theorem for the automorphism group of a smooth toric variety, previously considered quite inaccessible.