تقرير
Answers to questions about medial layer graphs of self-dual regular and chiral polytopes
العنوان: | Answers to questions about medial layer graphs of self-dual regular and chiral polytopes |
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المؤلفون: | Conder, Marston, Steinmann, Isabelle |
المصدر: | In: Ars Mathematica Contemporanea (Dec. 2023) |
سنة النشر: | 2024 |
المجموعة: | Mathematics |
مصطلحات موضوعية: | Mathematics - Combinatorics, 52B15, 05E18 |
الوصف: | An abstract $n$-polytope $\mathcal{P}$ is a partially-ordered set which captures important properties of a geometric polytope, for any dimension $n$. For even $n \ge 2$, the incidences between elements in the middle two layers of the Hasse diagram of $\mathcal{P}$ give rise to the medial layer graph of $\mathcal{P}$, denoted by $\mathcal{G} = \mathcal{G}(\mathcal{P})$. If $n=4$, and $\mathcal{P}$ is both highly symmetric and self-dual of type $\{p,q,p\}$, then a Cayley graph $\mathcal{C}$ covering $\mathcal{G}$ can be constructed on a group of polarities of $\mathcal{P}$. In this paper we address some open questions about the relationship between $\mathcal{G}$ and $\mathcal{C}$ that were raised in a 2008 paper by Monson and Weiss, and describe some interesting examples of these graphs. In particular, we give the first known examples of improperly self-dual chiral polytopes of type $\{3,q,3\}$, which are also among the very few known examples of highly symmetric self-dual finite polytopes that do not admit a polarity. Also we show that if $p=3$ then $\mathcal{C}$ cannot have a higher degree of $s$-arc-transitivity than $\mathcal{G}$, and we present a family of regular $4$-polytopes of type $\{6,q,6\}$ for which the vertex-stabilisers in the automorphism group of $\mathcal{C}$ are larger than those for $\mathcal{G}$. |
نوع الوثيقة: | Working Paper |
DOI: | 10.26493/1855-3974.3229.8b1 |
URL الوصول: | http://arxiv.org/abs/2406.13848 |
رقم الأكسشن: | edsarx.2406.13848 |
قاعدة البيانات: | arXiv |
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