restriction(Arrangement,Ideal) -- construct the restriction a hyperplane arrangement to a subspace

Synopsis

Function: restriction
Usage:

restriction(A, I)

restriction(A, F)

A ^ F

restriction F
Inputs:
- A, a hyperplane arrangement,
- I, an ideal, an ideal defining the subspace to which we restrict. One may also specify a single ring element or a set of indices. In the latter case, the subspace is the intersection of the corresponding hyperplanes.
Outputs:
- a hyperplane arrangement,

Description

The restriction of an arrangement ${\mathcal A}$ to a subspace $X$ is the (multi)arrangement with hyperplanes $H_i\cap X$, where $H\in {\mathcal A}$ but $H\not\supseteq X$. The subspace $X$ may be defined by a ring element or an ideal.

If an index or list (or set) of hyperplanes $S$ is given, then $X=\bigcap_{i\in S}H_i$. In this case, the restriction is a realization of the matroid contraction $M/S$, where $M$ denotes the matroid of ${\mathcal A}$.

In general, the restriction is denoted ${\mathcal A}^X$. Its ambient space is $X$.

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 = {{3, 4, 5}, {1, 2, 5}, {0, 5}, {0, 2, 4}, {1, 4}, {2, 3}, {0, 1, 3}}

o2 : List

i3 : A' = restriction first L

o3 = {x  - x , x  - x , x  - x }
       1    4   1    4   1    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

Unfortunately, the term ``restriction'' is used in conflicting senses in arrangements versus matroids literature. In the latter terminology, ``restriction'' to $S$ is a synonym for the deletion of the complement of $S$.

Ways to use this method:

restriction(Arrangement,Flat)
restriction(Arrangement,Ideal) -- construct the restriction a hyperplane arrangement to a subspace
restriction(Arrangement,List)
restriction(Arrangement,RingElement)
restriction(Arrangement,Set)
restriction(Arrangement,ZZ)
restriction(Flat)

restriction(Arrangement,Ideal) -- construct the restriction a hyperplane arrangement to a subspace

Synopsis

Description

See also

Ways to use this method: