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

Synopsis

• Function: matrix
• Usage:
matrix A
• Inputs:
• Optional inputs:
• Degree => ..., default value null, this optional input is ignored by this function
• Outputs:
• , 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)