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Abstract
Given an ascending chain (In)n∈N of Sym-invariant squarefree monomial ideals, we study the corresponding chain of Alexander duals (In∨)n∈N. Using a novel combinatorial tool, which we call avoidance up to symmetry, we provide an explicit description of the minimal generating set up to symmetry in terms of the original generators. Combining this result with methods from discrete geometry, this enables us to show that the number of orbit generators of In∨ is given by a polynomial in n for sufficiently large n. The same is true for the number of orbit generators of minimal degree, this degree being a linear function in n eventually. The former result implies that the number of Sym-orbits of primary components of In grows polynomially in n for large n. As another application, we show that, for each i ≥ 0, the number of i-dimensional faces of the associated Stanley-Reisner complexes of In is also given by a polynomial in n for large n.
Document Type
Article
Publication Date
2026
Digital Object Identifier (DOI)
https://doi.org/10.1016/j.aam.2026.103082
Funding Information
The first author was partially supported by the NSF GRFP under Grant No. DGE-1650441. The fourth author was partially supported by Simons Foundation grant #636513. The fifth author was partially supported by NSF grant DMS-2053288.
Repository Citation
Almousa, Ayah; Bruegge, Kaitlin; Juhnke-Kubitzke, Martina; Nagel, Uwe; and Pevzner, Alexandra, "Alexander duals of symmetric simplicial complexes and Stanley-Reisner ideals" (2026). Mathematics Faculty Publications. 51.
https://uknowledge.uky.edu/math_facpub/51
Notes/Citation Information
0196-8858/© 2026 Published by Elsevier Inc.