Characterization of Stanley-Reisner varieties by their automorphism group

We study the automorphism ind-group of a Stanley-Reisner variety $X_Δ$ through a combinatorial toolkit on the underlying complex: a facet closure operator, its Demazure roots, and the resulting dichotomy between exposed and hidden facets. Our main theorem is that a fully exposed complex, that is, one in which every facet has a private vertex, is recovered from the ind-group: $\mathrm{Aut}(X_Δ)\cong\mathrm{Aut}(X_{Δ'})$ forces $Δ\congΔ'$. The hypothesis cannot be dropped, but it holds after one stabilization, so for arbitrary $Δ,Δ'$ an isomorphism $\mathrm{Aut}(X_Δ\times\mathbb{A}^1)\cong\mathrm{Aut}(X_{Δ'}\times\mathbb{A}^1)$ already forces $Δ\congΔ'$. At the opposite extreme, a fully hidden $Δ$ gives $\mathrm{Aut}(X_Δ)=T_0\rtimes S(Δ)$, never isomorphic to the ind-group of a non-rigid Stanley-Reisner variety. The criterion decides graphs and skeleta, and every complex is homotopy equivalent to a fully exposed one.