matrix(Arrangement) -- make a matrix from the defining equations

Synopsis

Function: matrix
Usage:

matrix A
Inputs:
- A, a hyperplane arrangement,
Optional inputs:
- Degree => ..., default value null, this optional input is ignored by this function
Outputs:
- a matrix, having one row, whose entries are the defining equations

Description

A hyperplane arrangement is defined by a list of affine-linear equations. This methods creates a matrix, over the underlying ring of the hyperplane arrangement, whose entries are the defining equations.

A few reflection arrangements yield the following matrices.

i1 : A = typeA 3

o1 = {x  - x , x  - x , x  - x , x  - x , x  - x , x  - x }
       1    2   1    3   1    4   2    3   2    4   3    4

o1 : Hyperplane Arrangement

i2 : R = ring A

o2 = R

o2 : PolynomialRing

i3 : matrix A

o3 = | x_1-x_2 x_1-x_3 x_1-x_4 x_2-x_3 x_2-x_4 x_3-x_4 |

             1      6
o3 : Matrix R  <-- R

i4 : matrix typeB 2

o4 = | x_1 x_1-x_2 x_1+x_2 x_2 |

                        1                 4
o4 : Matrix (QQ[x ..x ])  <-- (QQ[x ..x ])
                 1   2             1   2

i5 : matrix typeD 4

o5 = | x_1-x_2 x_1+x_2 x_1-x_3 x_1+x_3 x_1-x_4 x_1+x_4 x_2-x_3 x_2+x_3
     ------------------------------------------------------------------------
     x_2-x_4 x_2+x_4 x_3-x_4 x_3+x_4 |

                        1                 12
o5 : Matrix (QQ[x ..x ])  <-- (QQ[x ..x ])
                 1   4             1   4

The trivial arrangement has no equations.

i6 : trivial = arrangement({},R)

o6 = {}

o6 : Hyperplane Arrangement

i7 : matrix trivial

o7 = 0

             1
o7 : Matrix R  <-- 0

i8 : assert(matrix trivial == 0)

Ways to use this method: