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# arrangement(Matrix,Ring) -- make a hyperplane arrangement

## Synopsis

• Function: arrangement
• Usage:
arrangement(M, R)
arrangement M
• Inputs:
• M, , a matrix whose columns represent linear forms defining hyperplanes
• R, a ring, a polynomial ring or linear quotient of a polynomial ring
• Outputs:
• , determined by the input data

## Description

A hyperplane is an affine-linear subspace of codimension one. An arrangement is a finite set of hyperplanes. When each hyperplane contains the origin, the arrangement is central.

Probably the best-known hyperplane arrangement is the braid arrangement consisting of all the diagonal hyperplanes. In $4$-space, it is constructed as follows.

 i1 : S = QQ[w,x,y,z]; i2 : A3 = arrangement(matrix{{1,1,1,0,0,0},{-1,0,0,1,1,0},{0,-1,0,-1,0,1},{0,0,-1,0,-1,-1}}, S) o2 = {w - x, w - y, w - z, x - y, x - z, y - z} o2 : Hyperplane Arrangement  i3 : assert isCentral A3

If we project along onto a subspace, then we obtain an essential arrangement, meaning that the rank of the arrangement is equal to the dimension of its ambient vector space.

 i4 : R = S/ideal(w+x+y+z); i5 : A3' = arrangement(matrix{{1,1,1,0,0,0},{-1,0,0,1,1,0},{0,-1,0,-1,0,1},{0,0,-1,0,-1,-1}}, R) o5 = {- 2x - y - z, - x - 2y - z, - x - y - 2z, x - y, x - z, y - z} o5 : Hyperplane Arrangement  i6 : ring A3' o6 = R o6 : QuotientRing i7 : assert(rank A3' === dim ring A3')

The trivial arrangement has no equations.

 i8 : trivial = arrangement(map(S^4,S^0,0),S) o8 = {} o8 : Hyperplane Arrangement  i9 : ring trivial o9 = S o9 : PolynomialRing i10 : assert isCentral trivial