next | previous | forward | backward | up | index | toc

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

## Synopsis

• Function: dual
• Usage:
dual A or dual(A, R)
• Inputs:
• Outputs:
• , 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)