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

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Abstract

In this paper, we introduce a fundamental substructure of maximal caps in the affine geometry AG(4, 3) that we call 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 \(S_6\).

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Notes

  1. The lines in a \(3\times 3\) subgrid, also called a plane, are three points horizontally, vertically, or diagonally, including diagonals as on a torus; the points in all other lines appear in three subgrids that are in the same position as a line in the \(3\times 3\) subgrids, so that, when the three subgrids are superimposed, those points either lie on top of one another, or they are in the same positions as a line in a subgrid. See Fig. 1.

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Funding

Research supported by NSF grant DMS-1063070.

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Authors and Affiliations

  1. Department of Statistics, Purdue University, West Lafayette, IN, USA

    Jordan Awan

  2. Mathematics Department, Boston College, Chestnut Hill, MA, USA

    Claire Frechette

  3. Willamette University, Salem, OR, USA

    Yumi Li

  4. Mathematics Department, Lafayette College, Easton, PA, USA

    Elizabeth McMahon

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  1. Jordan Awan
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  2. Claire Frechette
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  3. Yumi Li
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  4. Elizabeth McMahon
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Correspondence to Elizabeth McMahon.

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Awan, J., Frechette, C., Li, Y. et al. Demicaps in AG(4,3) and Maximal Cap Partitions. Graphs and Combinatorics 38, 193 (2022). https://doi.org/10.1007/s00373-022-02568-x

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  • DOI: https://doi.org/10.1007/s00373-022-02568-x

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