A partial order on the 240 packings of PG(3,2)
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909887437471744 |
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| author | Green, R. M. |
| author_facet | Green, R. M. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_03040 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A partial order on the 240 packings of PG(3,2) Green, R. M. Combinatorics Group Theory 05E18, 51E23, 06A07, 20F55 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. |
| title | A partial order on the 240 packings of PG(3,2) |
| topic | Combinatorics Group Theory 05E18, 51E23, 06A07, 20F55 |
| url | https://arxiv.org/abs/2511.03040 |