发表机构
Universidad de La Serena; Pontificia Universidad Católica de Chile（拉塞雷纳大学; 智利天主教 Pontifícia 大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文利用组合工具研究Stanley-Reisner簇的自同构ind群，证明完全外露复形可由其ind群唯一确定，稳定后任意复形的乘积簇也满足该性质，还给出完全隐藏复形的自同构群形式及应用准则。
AI 中文摘要
我们通过底层复形上的组合工具研究Stanley-Reisner簇$X_\Delta$的自同构ind群：包括面闭包算子、其Demazure根，以及由此产生的外露面与隐藏面的二分。我们的主要定理是，完全外露复形（即每个面都有一个私有顶点）可由ind群恢复：$\mathrm{Aut}(X_\Delta)\cong\mathrm{Aut}(X_{\Delta'})$蕴含$\Delta\cong\Delta'$。该假设不能省略，但在一次稳定后成立，因此对任意$\Delta,\Delta'$，同构$\mathrm{Aut}(X_\Delta\times\mathbb{A}^1)\cong\mathrm{Aut}(X_{\Delta'}\times\mathbb{A}^1)$已蕴含$\Delta\cong\Delta'$。在另一极端，完全隐藏的$\Delta$给出$\mathrm{Aut}(X_\Delta)=T_0\rtimes S(\Delta)$，其永远不与非刚性Stanley-Reisner簇的ind群同构。该准则可判定图与骨架，且每个复形同伦等价于一个完全外露复形。
英文摘要
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.
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