Combinatorial simpliciality of arrangements of hyperplanes

Abstract

We introduce a combinatorial characterization of simpliciality for arrangements of hyperplanes. We then give a sharp upper bound for the number of hyperplanes of such an arrangement in the projective plane over a finite field, and present some series of arrangements related to the known arrangements in characteristic zero. We further enumerate simplicial arrangements with given symmetry groups. Finally, we determine all finite complex reflection groups affording combinatorially simplicial arrangements. It turns out that combinatorial simpliciality coincides with inductive freeness for finite complex reflection groups except for the Shephard–Todd group $G_{31}$.

Appendix A: Some simplicial arrangements over \(\mathbb {C}\)

We reproduce here the newly found simplicial arrangements over $\mathbb {C}$ in the most compact notation, namely as sets of normal vectors in the original finite field. The ordering is the same as in 5.2. The symbol $\omega$ stands for a primitive element of the corresponding field. As in 5.2, we omit those incidences which come from real simplicial arrangements, from finite complex reflection groups, or which are near pencil.

$q=5$, $\,\, \{(0,1,1)$, $(1,0,0)$, $(1,0,1)$, $(1,0,2)$, $(1,1,0)$, $(1,2,1)$, $(1,3,2)$, $(1,4,0)$, $(1,4,1)$, $(1,4,2)$, $(1,4,3)$, $(1,4,4)\}$,

$q=7$, $\,\, \{(0,1,0)$, $(0,1,2)$, $(0,1,3)$, $(0,1,4)$, $(1,0,0)$, $(1,0,1)$, $(1,1,3)$, $(1,2,0)$, $(1,2,2)$, $(1,3,0)$, $(1,4,1)$, $(1,5,1)$, $(1,6,3)$, $(1,6,5)\}$,

$q=7$, $\,\, \{(0,1,0)$, $(0,1,2)$, $(0,1,3)$, $(0,1,4)$, $(1,0,0)$, $(1,0,1)$, $(1,0,5)$, $(1,1,3)$, $(1,2,0)$, $(1,2,2)$, $(1,3,0)$, $(1,4,1)$, $(1,5,1)$, $(1,6,3)$, $(1,6,5)\}$,

$q=7$, $\,\, \{(0,0,1)$, $(1,0,0)$, $(1,0,5)$, $(1,0,6)$, $(1,1,0)$, $(1,1,2)$, $(1,1,6)$, $(1,2,0)$, $(1,2,2)$, $(1,2,6)$, $(1,3,0)$, $(1,3,5)$, $(1,5,1)$, $(1,5,3)$, $(1,5,4)\}$,

$q=7$, $\,\, \{(0,1,2)$, $(1,0,0)$, $(1,0,2)$, $(1,1,4)$, $(1,1,5)$, $(1,1,6)$, $(1,2,3)$, $(1,3,0)$, $(1,3,2)$, $(1,3,4)$, $(1,4,0)$, $(1,4,1)$, $(1,4,4)$, $(1,5,6)$, $(1,6,6)\}$,

$q = 9$, $\,\, \{(0,0,1)$, $(1,0,0)$, $(1,0,\omega ^2)$, $(1,0,2)$, $(1,1,2)$, $(1,\omega ^3,2)$, $(1,\omega ^3,\omega ^6)$, $(1,2,\omega ^3)$, $(1,2,\omega ^5)$, $(1,\omega ^5,0)$, $(1,\omega ^5,1)$, $(1,\omega ^5,2)$, $(1,\omega ^6,0)$,

$(1,\omega ^6,\omega ^5)$, $(1,\omega ^6,\omega ^7)\}$,

$q=7$, $\,\, \{(0,1,2)$, $(1,0,0)$, $(1,1,4)$, $(1,1,5)$, $(1,1,6)$, $(1,2,2)$, $(1,2,3)$, $(1,3,0)$, $(1,3,4)$, $(1,4,0)$, $(1,4,1)$, $(1,4,4)$, $(1,5,6)$, $(1,6,2)$, $(1,6,6)\}$,

$q=7$, $\,\, \{(0,0,1)$, $(0,1,2)$, $(1,0,1)$, $(1,2,0)$, $(1,2,2)$, $(1,2,6)$, $(1,3,0)$, $(1,4,0)$, $(1,4,1)$, $(1,4,2)$, $(1,5,0)$, $(1,5,2)$, $(1,5,3)$, $(1,6,2)$, $(1,6,4)$, $(1,6,5)\}$,

$q=7$, $\,\, \{(0,0,1)$, $(0,1,2)$, $(1,0,0)$, $(1,0,1)$, $(1,2,0)$, $(1,2,2)$, $(1,2,6)$, $(1,3,0)$, $(1,4,0)$, $(1,4,1)$, $(1,4,2)$, $(1,5,0)$, $(1,5,2)$, $(1,5,3)$, $(1,6,2)$, $(1,6,4)$, $(1,6,5)\}$,

$q=7$, $\,\, \{(0,0,1)$, $(0,1,2)$, $(1,0,0)$, $(1,0,1)$, $(1,0,2)$, $(1,1,4)$, $(1,1,6)$, $(1,2,2)$, $(1,2,6)$, $(1,3,0)$, $(1,3,2)$, $(1,3,4)$, $(1,4,0)$, $(1,4,1)$, $(1,5,6)$, $(1,6,2)$, $(1,6,4)\}$,

$q=7$, $\,\, \{(0,0,1)$, $(0,1,0)$, $(0,1,3)$, $(0,1,4)$, $(1,0,0)$, $(1,0,1)$, $(1,0,2)$, $(1,1,0)$, $(1,1,1)$, $(1,1,3)$, $(1,2,0)$, $(1,2,4)$, $(1,2,5)$, $(1,3,0)$, $(1,4,1)$, $(1,4,5)$, $(1,4,6)$, $(1,6,5)\}$,

$q=7$, $\,\, \{(0,0,1)$, $(0,1,0)$, $(1,0,0)$, $(1,0,2)$, $(1,0,5)$, $(1,0,6)$, $(1,1,0)$, $(1,1,2)$, $(1,1,6)$, $(1,2,0)$, $(1,2,2)$, $(1,2,6)$, $(1,3,0)$, $(1,3,5)$, $(1,4,5)$, $(1,5,1)$, $(1,5,3)$, $(1,5,4)\}$,

$q = 9$, $\,\, \{(0,0,1)$, $(1,0,0)$, $(1,0,\omega )$, $(1,0,\omega ^2)$, $(1,0,2)$, $(1,1,2)$, $(1,1,\omega ^7)$, $(1,\omega ^3,2)$, $(1,\omega ^3,\omega ^6)$, $(1,2,\omega ^3)$, $(1,2,\omega ^5)$, $(1,\omega ^5,0)$, $(1,\omega ^5,1)$, $(1,\omega ^5,2)$,

$(1,\omega ^6,0)$,

$(1,\omega ^6,2)$, $(1,\omega ^6,\omega ^5)$, $(1,\omega ^6,\omega ^7)\}$,

$q=7$, $\,\, \{(0,1,0)$, $(0,1,1)$, $(0,1,3)$, $(1,0,3)$, $(1,0,5)$, $(1,0,6)$, $(1,1,1)$, $(1,1,5)$, $(1,3,3)$, $(1,3,6)$, $(1,4,5)$, $(1,4,6)$, $(1,5,0)$, $(1,5,2)$, $(1,5,3)$, $(1,6,1)$, $(1,6,2)$, $(1,6,4)\}$,

$q=7$, $\,\, \{(0,0,1)$, $(0,1,1)$, $(0,1,2)$, $(1,0,1)$, $(1,0,2)$, $(1,1,1)$, $(1,1,4)$, $(1,1,6)$, $(1,2,6)$, $(1,3,0)$, $(1,3,2)$, $(1,3,4)$, $(1,4,0)$, $(1,4,1)$, $(1,4,6)$, $(1,5,4)$, $(1,5,6)$, $(1,6,4)\}$,

$q = 9$, $\,\, \{(0,0,1)$, $(1,0,0)$, $(1,0,\omega ^2)$, $(1,0,2)$, $(1,1,2)$, $(1,1,\omega ^6)$, $(1,\omega ^3,2)$, $(1,\omega ^3,\omega ^6)$, $(1,\omega ^3,\omega ^7)$, $(1,2,\omega ^3)$, $(1,2,\omega ^5)$, $(1,2,\omega ^6)$, $(1,\omega ^5,0)$, $(1,\omega ^5,1)$,

$(1,\omega ^5,2)$, $(1,\omega ^6,0)$, $(1,\omega ^6,\omega ^5)$, $(1,\omega ^6,\omega ^7)\}$,

$q=7$, $\,\, \{(0,0,1)$, $(0,1,0)$, $(0,1,3)$, $(0,1,4)$, $(1,0,0)$, $(1,0,1)$, $(1,0,2)$, $(1,0,5)$, $(1,1,0)$, $(1,1,1)$, $(1,1,3)$, $(1,2,0)$, $(1,2,4)$, $(1,2,5)$, $(1,3,0)$, $(1,4,1)$, $(1,4,5)$, $(1,4,6)$, $(1,6,5)\}$,

$q = 9$, $\,\, \{(0,1,1)$, $(1,0,\omega )$, $(1,1,0)$, $(1,1,\omega ^2)$, $(1,1,\omega ^7)$, $(1,\omega ,1)$, $(1,\omega ,\omega ^7)$, $(1,\omega ^2,\omega ^6)$, $(1,\omega ^3,\omega )$, $(1,\omega ^3,\omega ^3)$, $(1,2,\omega ^3)$, $(1,2,\omega ^7)$, $(1,\omega ^5,1)$, $(1,\omega ^6,\omega ^7)$, $(1,\omega ^7,0)$, $(1,\omega ^7,2)$, $(1,\omega ^7,\omega ^5)$, $(1,\omega ^7,\omega ^6)$, $(1,\omega ^7,\omega ^7)\}$,

$q = 9$, $\,\, \{(0,1,\omega ^5)$, $(1,0,0)$, $(1,0,\omega )$, $(1,0,\omega ^2)$, $(1,0,2)$, $(1,1,0)$, $(1,1,1)$, $(1,1,2)$, $(1,1,\omega ^5)$, $(1,1,\omega ^7)$, $(1,\omega ,\omega ^7)$, $(1,\omega ^2,\omega ^5)$, $(1,\omega ^5,1)$, $(1,\omega ^5,2)$, $(1,\omega ^6,0)$,

$(1,\omega ^6,2)$, $(1,\omega ^6,\omega ^5)$, $(1,\omega ^6,\omega ^7)$, $(1,\omega ^7,\omega ^3)\}$,

$q = 9$, $\,\, \{(0,1,1)$, $(1,0,\omega )$, $(1,1,0)$, $(1,1,\omega ^2)$, $(1,1,\omega ^7)$, $(1,\omega ,1)$, $(1,\omega ,\omega ^7)$, $(1,\omega ^3,\omega )$, $(1,\omega ^3,\omega ^3)$, $(1,\omega ^3,\omega ^6)$, $(1,2,\omega ^3)$, $(1,2,\omega ^6)$, $(1,2,\omega ^7)$, $(1,\omega ^5,1)$, $(1,\omega ^6,\omega ^7)$, $(1,\omega ^7,0)$, $(1,\omega ^7,2)$, $(1,\omega ^7,\omega ^5)$, $(1,\omega ^7,\omega ^7)\}$,

$q=7$, $\,\, \{(0,0,1)$, $(0,1,2)$, $(1,0,1)$, $(1,0,4)$, $(1,0,6)$, $(1,1,3)$, $(1,2,0)$, $(1,2,2)$, $(1,2,6)$, $(1,3,0)$, $(1,4,0)$, $(1,4,1)$, $(1,4,2)$, $(1,4,5)$, $(1,5,0)$, $(1,5,2)$, $(1,5,3)$, $(1,6,2)$, $(1,6,4)$, $(1,6,5)\}$,

$q=13$, $\,\, \{(0,1,10)$, $(1,0,0)$, $(1,0,3)$, $(1,0,4)$, $(1,1,2)$, $(1,2,8)$, $(1,3,0)$, $(1,4,2)$, $(1,4,7)$, $(1,4,9)$, $(1,5,8)$, $(1,6,0)$, $(1,8,1)$, $(1,9,2)$, $(1,9,5)$, $(1,9,9)$, $(1,9,11)$, $(1,10,1)$, $(1,10,5)$, $(1,10,9)$, $(1,11,1)\}$,

$q = 9$, $\,\, \{(0,0,1)$, $(0,1,1)$, $(0,1,\omega )$, $(0,1,\omega ^3)$, $(0,1,2)$, $(0,1,\omega ^6)$, $(1,1,0)$, $(1,1,\omega ^7)$, $(1,\omega ,0)$, $(1,\omega ,1)$, $(1,\omega ^2,0)$, $(1,\omega ^2,\omega )$, $(1,\omega ^3,0)$, $(1,\omega ^3,\omega ^2)$, $(1,2,0)$, $(1,2,\omega ^3)$, $(1,\omega ^5,0)$, $(1,\omega ^5,2)$, $(1,\omega ^6,0)$, $(1,\omega ^6,\omega ^5)$, $(1,\omega ^7,0)$, $(1,\omega ^7,\omega ^6)\}$,

$q = 9$, $\,\, \{(0,1,1)$, $(0,1,\omega )$, $(0,1,\omega ^3)$, $(0,1,2)$, $(0,1,\omega ^5)$, $(0,1,\omega ^7)$, $(1,1,\omega ^2)$, $(1,1,\omega ^6)$, $(1,\omega ,\omega ^3)$, $(1,\omega ,\omega ^7)$, $(1,\omega ^2,1)$, $(1,\omega ^2,2)$, $(1,\omega ^3,\omega )$, $(1,\omega ^3,\omega ^5)$,

$(1,2,\omega ^2)$, $(1,2,\omega ^6)$, $(1,\omega ^5,\omega ^3)$, $(1,\omega ^5,\omega ^7)$, $(1,\omega ^6,1)$, $(1,\omega ^6,2)$, $(1,\omega ^7,\omega )$, $(1,\omega ^7,\omega ^5)\}$,

$q = 9$, $\,\, \{(0,0,1)$, $(0,1,0)$, $(0,1,1)$, $(0,1,\omega )$, $(0,1,2)$, $(0,1,\omega ^7)$, $(1,1,\omega ^2)$, $(1,1,\omega ^6)$, $(1,\omega ,\omega ^3)$, $(1,\omega ,\omega ^7)$, $(1,\omega ^2,1)$, $(1,\omega ^2,2)$, $(1,\omega ^3,\omega )$, $(1,\omega ^3,\omega ^5)$,

$(1,2,\omega ^2)$, $(1,2,\omega ^6)$, $(1,\omega ^5,\omega ^3)$, $(1,\omega ^5,\omega ^7)$, $(1,\omega ^6,1)$, $(1,\omega ^6,2)$, $(1,\omega ^7,\omega )$, $(1,\omega ^7,\omega ^5)\}$,

$q = 9$, $\,\, \{(0,0,1)$, $(0,1,0)$, $(0,1,\omega ^2)$, $(0,1,2)$, $(0,1,\omega ^6)$, $(1,0,0)$, $(1,1,\omega ^3)$, $(1,1,\omega ^7)$, $(1,\omega ,\omega ^2)$, $(1,\omega ,\omega ^6)$, $(1,\omega ^2,\omega )$, $(1,\omega ^2,\omega ^5)$, $(1,\omega ^3,1)$, $(1,\omega ^3,2)$,

$(1,2,\omega ^3)$, $(1,2,\omega ^7)$, $(1,\omega ^5,\omega ^2)$, $(1,\omega ^5,\omega ^6)$, $(1,\omega ^6,\omega )$, $(1,\omega ^6,\omega ^5)$, $(1,\omega ^7,1)$, $(1,\omega ^7,2)\}$,

$q=11$, $\,\, \{(0,1,3)$, $(0,1,7)$, $(1,0,4)$, $(1,0,6)$, $(1,1,4)$, $(1,1,6)$, $(1,1,9)$, $(1,1,10)$, $(1,2,4)$, $(1,2,6)$, $(1,2,8)$, $(1,3,4)$, $(1,3,10)$, $(1,4,0)$, $(1,4,4)$, $(1,4,10)$, $(1,5,6)$, $(1,6,2)$, $(1,6,3)$, $(1,6,5)$, $(1,7,10)$, $(1,8,4)$, $(1,9,0)$, $(1,9,1)$, $(1,10,7)\}$,

$q=13$, $\,\, \{(0,1,9)$, $(1,0,4)$, $(1,1,10)$, $(1,1,11)$, $(1,1,12)$, $(1,2,7)$, $(1,2,8)$, $(1,2,11)$, $(1,3,7)$, $(1,4,7)$, $(1,5,5)$, $(1,6,0)$, $(1,6,6)$, $(1,6,8)$, $(1,7,6)$, $(1,7,8)$, $(1,7,10)$, $(1,8,5)$, $(1,9,4)$, $(1,9,9)$, $(1,9,12)$, $(1,10,5)$, $(1,10,11)$, $(1,10,12)$, $(1,11,0)$, $(1,11,3)$, $(1,11,10)$, $(1,12,0)$, $(1,12,3)$, $(1,12,6)\}$,

$q=13$, $\,\, \{(0,1,7)$, $(1,0,4)$, $(1,0,6)$, $(1,0,9)$, $(1,1,2)$, $(1,2,5)$, $(1,2,8)$, $(1,3,8)$, $(1,4,3)$, $(1,4,7)$, $(1,4,10)$, $(1,5,3)$, $(1,6,0)$, $(1,6,5)$, $(1,7,7)$, $(1,7,10)$, $(1,7,12)$, $(1,8,12)$, $(1,9,0)$, $(1,9,4)$, $(1,9,6)$, $(1,9,9)$, $(1,9,11)$, $(1,9,12)$, $(1,10,0)$, $(1,10,5)$, $(1,10,10)$, $(1,11,1)$, $(1,12,6)$, $(1,12,8)$, $(1,12,9)\}$,

$q=29$, $\,\, \{(0,1,6)$, $(0,1,23)$, $(1,0,0)$, $(1,0,1)$, $(1,0,2)$, $(1,1,0)$, $(1,1,1)$, $(1,1,22)$, $(1,2,0)$, $(1,2,23)$, $(1,2,26)$, $(1,3,22)$, $(1,3,23)$, $(1,3,26)$, $(1,4,15)$, $(1,5,6)$, $(1,5,7)$, $(1,5,21)$, $(1,6,21)$, $(1,7,12)$, $(1,7,13)$, $(1,7,19)$, $(1,8,5)$, $(1,10,9)$, $(1,10,12)$, $(1,10,15)$, $(1,11,0)$, $(1,11,2)$, $(1,11,12)$, $(1,12,1)$, $(1,13,2)$,

$(1,13,4)$, $(1,13,10)$, $(1,13,27)$, $(1,15,15)$, $(1,16,8)$, $(1,17,23)$, $(1,19,28)$, $(1,20,16)$, $(1,22,11)$, $(1,25,20)$, $(1,25,22)$, $(1,25,24)$, $(1,26,3)$, $(1,26,21)\}$.