Year: 2024
Type: article
Abstract: In 1999 Michael Hartley showed that any abstract polytope can be constructed as a double coset poset, by means of a C-group $$\mathcal {W}$$ W and a subgroup $$N \le \mathcal {W}$$ N ≤ W . Subgroups $$N \le \mathcal {W}$$ N ≤ W that give rise to abstract polytopes through such a construction are called sparse. If, further, the stabilizer of a base flag of the poset is precisely N, then N is said to be semisparse. In [4, Conjecture 5.2] Hartely conjectures that sparse groups are always semisparse. In this paper, we show that this conjecture is in fact false: there exist sparse groups that are not semisparse. In particular, we show that such groups are always obtained from non-faithful maniplexes that give rise to polytopes. Using this, we show that Hartely’s conjecture holds for rank 3, but we construct examples to disprove the conjecture for all ranks $$n\ge 4$$ n ≥ 4 .
Source: Aequationes Mathematicae
Authors: Isabel Hubard, Micael Toledo
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,990
Awards: G622329931, G6740841446
OpenAlex ID: W4404740499