Characterization of Stanley-Reisner varieties by their automorphism group
Abstract
We study the automorphism ind-group of a Stanley-Reisner variety XΔ 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: Aut(XΔ)≅Aut(XΔ') forces Δ≅Δ'. The hypothesis cannot be dropped, but it holds after one stabilization, so for arbitrary Δ,Δ' an isomorphism Aut(XΔ×𝔸1)≅Aut(XΔ'×𝔸1) already forces Δ≅Δ'. At the opposite extreme, a fully hidden Δ gives Aut(XΔ)=T0rtimes 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.
Related
- Stanley-Reisner Theory in Mixed Characteristic
- Global vector fields and the automorphism group of regular semisimple Hessenberg varieties
- Cube-freeness and a square formula for canonical trace ideals of Stanley--Reisner rings
- On orbits of the automorphism group of toroidal spherical varieties
- The Negami Polynomial and Broken-Circuit Stanley--Reisner Rings
BibTeX
@misc{diaz2026characterization,
title = {{Characterization of Stanley-Reisner varieties by their automorphism group}},
author = {Roberto Díaz and José Alejandro Samper},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02785},
eprint = {2609.02785},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02785},
}
RIS
TY - UNPB
TI - Characterization of Stanley-Reisner varieties by their automorphism group
AU - Roberto Díaz
AU - José Alejandro Samper
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02785
UR - https://arxiv.org/abs/2609.02785
AB - 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: Aut(X_Δ)≅Aut(X_Δ') forces Δ≅Δ'. The hypothesis cannot be dropped, but it holds after one stabilization, so for arbitrary Δ,Δ' an isomorphism Aut(X_Δ×𝔸^1)≅Aut(X_Δ'×𝔸^1) already forces Δ≅Δ'. At the opposite extreme, a fully hidden Δ gives Aut(X_Δ)=T_0rtimes 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.
ER -