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

TL;DR

This paper introduces demicaps, a fundamental substructure in affine geometry AG(4,3), revealing their role in partitioning maximal caps and connecting to affine group symmetries, including automorphisms of S_6.

Contribution

It defines demicaps in AG(4,3) and demonstrates their use in understanding maximal cap partitions and affine group actions, including automorphisms of S_6.

Findings

Demicaps form a key substructure linking maximal caps and partitions.

The collection of 36 maximal caps can be expressed via unions of disjoint demicaps.

Affine group actions include symmetries related to automorphisms of S_6.

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 $S_6$.