Demicaps in AG(4,3) and Their Relation to Maximal Cap Partitions

Jordan Awan; Clare Frechette; Yumi Li; Elizabeth McMahon

Demicaps in AG(4,3) and Their Relation to Maximal Cap Partitions

Combinatorics 2021-06-29 v1

Authors: Jordan Awan , Clare Frechette , Yumi Li , Elizabeth McMahon

Abstract

In this paper, we introduce a fundamental substructure of maximal caps in the affine geometry $AG(4,3)$ that we call \emph{demicaps}. Demicaps provide a direct link to particular partitions of $AG(4,3)$ into 4 maximal caps plus a single point. The full collection of 36 maximal caps that are in exactly one partition with a given cap $C$ can be expressed as unions of two disjoint demicaps taken from a set of 12 demicaps; these 12 can also be found using demicaps in $C$ . The action of the affine group on these 36 maximal caps includes actions related to the outer automorphisms of $S_6$ .

Keywords

affine geometry maximal caps demicaps cap partitions affine group outer automorphisms of S6

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@article{arxiv.2106.14141,
  title  = {Demicaps in AG(4,3) and Their Relation to Maximal Cap Partitions},
  author = {Jordan Awan and Clare Frechette and Yumi Li and Elizabeth McMahon},
  journal= {arXiv preprint arXiv:2106.14141},
  year   = {2021}
}

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19 pages, 16 figures

Demicaps in AG(4,3) and Their Relation to Maximal Cap Partitions

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