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

Synopsis

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)

See also

Ways to use this method: