# restriction -- restriction of arrangement to flat/hyperplane

## Synopsis

• Usage:
restriction(A,F) or restriction(A,x) or restriction(A,I)
• Inputs:
• A, , a hyperplane arrangement (optional)
• F, , flat to which you restrict
• x, , equation of hyperplane to which you restrict
• I, an ideal, an ideal defining a subspace to which you restrict
• Outputs:
• , the restriction of A

## Description

The restriction of an arrangement to the subspace X indexed by a flat is the (multi)set of hyperplanes H intersect X for all H in the arrangement A. In the first case, one can also write A^F.
 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 : L := flats(2,A) o2 = {{0, 1, 3}, {0, 2, 4}, {0, 5}, {1, 4}, {1, 2, 5}, {2, 3}, {3, 4, 5}} o2 : List i3 : A' := restriction first L o3 = {x - x , x - x , x - x } 3 4 3 4 3 4 o3 : Hyperplane Arrangement  i4 : x := (ring A)_0 -- the subspace need not be in the arrangement o4 = x 1 o4 : QQ[x ..x ] 1 4 i5 : restriction(A,x) o5 = {-x , -x , -x , x - x , x - x , x - x } 2 3 4 2 3 2 4 3 4 o5 : Hyperplane Arrangement 
The restriction is, in general, a multiarrangement. Use trim to eliminate repeated hyperplanes. For example,
 i6 : trim A' o6 = {x - x } 3 4 o6 : Hyperplane Arrangement