Sparse groups need not be semisparse
Por:
Hubard, I, Toledo, M
Publicada:
1 feb 2025
Ahead of Print:
1 nov 2024
Resumen:
In 1999 Michael Hartley showed that any abstract polytope can be constructed as a double coset poset, by means of a C-group W\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {W}$$\end{document} and a subgroup N <= W\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \le \mathcal {W}$$\end{document}. Subgroups N <= W\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \le \mathcal {W}$$\end{document} 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 >= 4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 4$$\end{document}.
Filiaciones:
Hubard, I:
Univ Nacl Autonoma Mexico, Inst Matemat, Mexico City 04510, Mexico
Toledo, M:
Univ Libre Bruxelles, Dept Math, CP216 Algebre & Combinatoire, Blvd Triomphe, B-1050 Brussels, Belgium
hybrid, All Open Access; Hybrid Gold Open Access
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