A partial order on the 240 packings of PG(3,2)

TL;DR

The paper constructs a canonical Bruhat-like, graded partial order on the 240 packings of $PG(3,2)$ by leveraging a quasiparabolic framework connected to the $E_8$ root system. It establishes a Lehmer-like code that realises these packings as a labellable element of a product of chains and proves a refined order compatible with signed permutation actions of type $D$, including faithful actions for parabolic subgroups. A robust correspondence is developed among packings, maximal totally singular subspaces of ${f F}_2^6$, and maximal orthogonal subsets of $E_8$, with explicit realizations via labelled Fano planes and a signed Fano-plane model. The results reveal deep structural links between finite geometry, root systems, Hecke algebras, and combinatorial topology, and provide a canonical labelling and symmetry framework for Kirkman-type configurations arising from $PG(3,2)$.

Abstract

It has long been known that the most symmetrical solutions of Kirkman's Schoolgirl Problem can be constructed from the $240$ packings of the projective space $PG(3, 2)$, but it seems to have escaped notice that these packings have the structure of a partially ordered set. In this paper, we construct a shellable Bruhat-like graded partial order on the packings of $PG(3, 2)$ that refines the partial order on the product of four chains $[8]\times[5]\times[3]\times[2]$ and defines a Lehmer code on the packings. The partial order exists because the packings of $PG(3, 2)$ form a quasiparabolic set (in the sense of Rains--Vazirani) that is in bijective correspondence with a certain collection of maximal orthogonal subsets of the $E_8$ root system. The $E_8$ construction also induces transitive actions of the Weyl groups of type $D_n$ on the packings for $5 \leq n \leq 8$, and these actions are faithful for $n < 8$. It is possible to define both the signed permutation action and the partial order using the combinatorics of labelled Fano planes.

Figures (7)

• Figure 1: The inequivalent labellings of the Fano plane corresponding to $x_0$ and $x_1$
• Figure 2: Dynkin diagram of type $E_8$
• Figure 3: The labelled Fano plane from Example \ref{['exa:oddeven']}
• Figure 4: The signed ${\mathbf 7}$-labelled Fano plane corresponding to $x_1$
• Figure 5: Signed ${\mathbf 6}$-labelled Pasch configuration corresponding to $x_1$

Theorems & Definitions (62)

• Example 2.1
• Example 2.2
• Definition 2.3
• Remark 2.4
• Definition 2.5
• Lemma 2.6
• proof
• Definition 2.7
• Remark 2.8
• proof