saliola / cn_hyperarr

Computation of congruence normality for hyperplane arrangements

Congruence Normality for Hyperplane Arrangements

This package is a SageMath package for computing congruence normality of rank-three, simplicial hyperplane arrangements.

This package includes a database of known rank-three simplicial hyperplane arrangements. It also includes modules for creating vector configurations and the three infinite families of simplicial rank-three arrangements. A vector configuration can be seen as the set of normals to a hyperplane arrangement. A simplicial hyperplane arrangement has a lattice of regions associated to each chamber. This lattice is congruence normal if it is obtainable through a sequence of doublings of convex sets. A hyperplane arrangement can be always or sometimes or never congruence normal, depending on whether its lattices of regions are congruence normal.

Here are examples of arrangements that are always, sometimes, and never congruence normal. First we load the normals of the three arrangements from the database. The entries of the database are labeled in the same way as in [CEL]:

sage: from cn_hyperarr import *
sage: always_normals = db_normals_CEL[(6,24,1)]
sage: somet_normals = db_normals_CEL[(10,60,3)]
sage: never_normals = db_normals_CEL[(22,288,1)]

Now we make them into vector configurations::

sage: always_vc = VectorConfiguration([vector(x) for x in always_normals])
sage: somet_vc = VectorConfiguration([vector(x) for x in somet_normals])
sage: never_vc = VectorConfiguration([vector(x) for x in never_normals])

To test congruence normality, use the function `RegionsCongruenceNormal`::

sage: always_check = RegionsCongruenceNormality(always_vc)
sage: always_vals_list = list(always_check.values())
sage: [always_vals_list.count(True), always_vals_list.count(False)]
[24,0]
sage: somet_check = RegionsCongruenceNormality(somet_vc)
sage: somet_vals_list = list(somet_check.values())
sage: [somet_vals_list.count(True), somet_vals_list.count(False)]
[40,20]
sage: never_check = RegionsCongruenceNormality(never_vc)
sage: never_vals_list = list(never_check.values())
sage: [never_vals_list.count(True), never_vals_list.count(False)]
[0,288]

The full documentation for the package can be found at https://sophiasage.github.io/cn_hyperarr/doc/html/

Notebooks

You can experience this package on a binder notebook (click on the binder image):

References

[CEL]

Michael Cuntz, Sophia Elia, and Jean-Philippe Labbé. Congruence normality of simplicial hyperplane arrangements via oriented matroids, 2020. arXiv:2009.14152.

[Gru]

Branko Grunbaum. A catalogue of simplicial arrangements in the real projective plane, 2009. Ars Math. Contemp. 2, no. 1, 1-25.

Installation

Local install from PyPi

Option 1: To install the package use the following command:

$ sage -pip install --user --upgrade cn-hyperarr

Local install from source

Option 2: To download the source from the git repository:

$ git clone https://github.com/sophiasage/cn_hyperarr.git

Change to the root directory and run:

$ sage -pip install --upgrade --no-index -v .

For convenience this package contains a [makefile](makefile) with this and other often used commands. Should you wish too, you can use the shorthand:

$ make install