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# 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:
• 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:

## 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 $S$.

• deletion -- deletion of subset of matroid
• subArrangement -- create the hyperplane arrangement containing a flat
• eulerRestriction -- form the Euler restriction of a central multiarrangement

## 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)