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HyperplaneArrangements :: restriction

restriction -- restriction of arrangement to flat/hyperplane

Synopsis

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 

See also

Ways to use restriction :

For the programmer

The object restriction is a method function.