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

TL;DR

This paper introduces a new partial order structure on the 240 packings of PG(3,2), linking combinatorics, root systems, and symmetry groups, revealing new algebraic and geometric insights.

Contribution

It constructs a shellable graded partial order on PG(3,2) packings, connecting them with E8 root systems and Weyl group actions, and provides a combinatorial framework using labelled Fano planes.

Findings

A shellable Bruhat-like partial order on PG(3,2) packings.

A bijection between packings and maximal orthogonal subsets of E8.

Faithful Weyl group actions on the packings for n<8.

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.