dual(CentralArrangement,Ring) -- the Gale dual of an arrangement

Synopsis

Function: dual
Usage:

dual A or dual(A, R)
Inputs:
- A, a central hyperplane arrangement,
- R, a ring,
Outputs:
- a central hyperplane arrangement, the Gale dual of A, optionally over the polynomial ring R.

Description

The dual of an arrangement of rank $r$ with $n$ hyperplanes is an arrangement of rank $n-r$ with $n$ hyperplanes, given by a linear realization of the dual matroid to that of ${\mathcal A}$. It is computed from a presentation of the kernel of the coefficient matrix of ${\mathcal A}$. If ${\mathcal A}$ is the arrangement of a planar graph then the dual of ${\mathcal A}$ is the arrangement of the dual graph.

i1 : A = arrangement "X2"

o1 = {x , x , x , x  - x , x  - x , x  + x , x  + x  - 2x }
       1   2   3   2    3   1    3   1    2   1    2     3

o1 : Hyperplane Arrangement

i2 : coefficients A

o2 = | 1 0 0 0  1  1 1  |
     | 0 1 0 1  0  1 1  |
     | 0 0 1 -1 -1 0 -2 |

              3       7
o2 : Matrix QQ  <-- QQ

i3 : A' = dual A

o3 = {- x  - x  - x , - x  - x  - x , x  + x  + 2x , x , x , x , x }
         2    3    4     1    3    4   1    2     4   1   2   3   4

o3 : Hyperplane Arrangement

i4 : coefficients dual A

o4 = | 0  -1 1 1 0 0 0 |
     | -1 0  1 0 1 0 0 |
     | -1 -1 0 0 0 1 0 |
     | -1 -1 2 0 0 0 1 |

              4       7
o4 : Matrix QQ  <-- QQ

i5 : assert (dual matroid A == matroid dual A)

Ways to use this method: