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# Arrangement -- the class of all hyperplane arrangements

## Description

A hyperplane is an affine-linear subspace of codimension one. An arrangement is a finite set of hyperplanes.

## Types of hyperplane arrangement:

- CentralArrangement -- the class of all central hyperplane arrangements

## Functions and methods returning a hyperplane arrangement:

- Arrangement ** Ring -- change the coefficient ring of an arrangement
- arrangement(Flat) -- get the hyperplane arrangement to which a flat belongs
- `arrangement` -- see arrangement(List,Ring) -- make a hyperplane arrangement
- `arrangement(List)` -- see arrangement(List,Ring) -- make a hyperplane arrangement
- arrangement(List,Ring) -- make a hyperplane arrangement
- `arrangement(RingElement)` -- see arrangement(List,Ring) -- make a hyperplane arrangement
- `arrangement(Matrix)` -- see arrangement(Matrix,Ring) -- make a hyperplane arrangement
- arrangement(Matrix,Ring) -- make a hyperplane arrangement
- `arrangement(String)` -- see arrangement(String,Ring) -- access a database of classic hyperplane arrangements
- `arrangement(String,PolynomialRing)` -- see arrangement(String,Ring) -- access a database of classic hyperplane arrangements
- arrangement(String,Ring) -- access a database of classic hyperplane arrangements
- `Arrangement ++ Arrangement` -- see arrangementSum(Arrangement,Arrangement) -- make the direct sum of two arrangements
- arrangementSum(Arrangement,Arrangement) -- make the direct sum of two arrangements
- compress(Arrangement) -- extract nonzero equations
- deCone(CentralArrangement,RingElement) -- produce an affine arrangement from a central one
- `deCone(CentralArrangement,ZZ)` -- see deCone(CentralArrangement,RingElement) -- produce an affine arrangement from a central one
- `deletion(Arrangement,List)` -- see deletion(Arrangement,RingElement) -- deletion of a subset of an arrangement
- deletion(Arrangement,RingElement) -- deletion of a subset of an arrangement
- `deletion(Arrangement,Set)` -- see deletion(Arrangement,RingElement) -- deletion of a subset of an arrangement
- `deletion(Arrangement,ZZ)` -- see deletion(Arrangement,RingElement) -- deletion of a subset of an arrangement
- `genericArrangement` -- see genericArrangement(ZZ,ZZ,Ring) -- realize the uniform matroid using points on the monomial curve
- `genericArrangement(ZZ,ZZ)` -- see genericArrangement(ZZ,ZZ,Ring) -- realize the uniform matroid using points on the monomial curve
- genericArrangement(ZZ,ZZ,Ring) -- realize the uniform matroid using points on the monomial curve
- `graphic(List)` -- see graphic(List,List,PolynomialRing) -- make a graphic arrangement
- `graphic(List,PolynomialRing)` -- see graphic(List,List,PolynomialRing) -- make a graphic arrangement
- `graphic(List,Ring)` -- see graphic(List,List,PolynomialRing) -- make a graphic arrangement
- prune(Arrangement) -- makes a new hyperplane arrangement in a polynomial ring
- randomArrangement(ZZ,PolynomialRing,ZZ) -- generate an arrangement at random
- `randomArrangement(ZZ,ZZ,ZZ)` -- see randomArrangement(ZZ,PolynomialRing,ZZ) -- generate an arrangement at random
- `Arrangement ^ Flat` -- see restriction(Arrangement,Ideal) -- construct the restriction a hyperplane arrangement to a subspace
- `restriction(Arrangement,Flat)` -- see restriction(Arrangement,Ideal) -- construct the restriction a hyperplane arrangement to a subspace
- restriction(Arrangement,Ideal) -- construct the restriction a hyperplane arrangement to a subspace
- `restriction(Arrangement,List)` -- see restriction(Arrangement,Ideal) -- construct the restriction a hyperplane arrangement to a subspace
- `restriction(Arrangement,RingElement)` -- see restriction(Arrangement,Ideal) -- construct the restriction a hyperplane arrangement to a subspace
- `restriction(Arrangement,Set)` -- see restriction(Arrangement,Ideal) -- construct the restriction a hyperplane arrangement to a subspace
- `restriction(Arrangement,ZZ)` -- see restriction(Arrangement,Ideal) -- construct the restriction a hyperplane arrangement to a subspace
- `restriction(Flat)` -- see restriction(Arrangement,Ideal) -- construct the restriction a hyperplane arrangement to a subspace
- `Arrangement _ Flat` -- see subArrangement(Arrangement,Flat) -- create the hyperplane arrangement containing a flat
- `subArrangement` -- see subArrangement(Arrangement,Flat) -- create the hyperplane arrangement containing a flat
- subArrangement(Arrangement,Flat) -- create the hyperplane arrangement containing a flat
- `subArrangement(Flat)` -- see subArrangement(Arrangement,Flat) -- create the hyperplane arrangement containing a flat
- `Arrangement ** RingMap` -- see substitute(Arrangement,RingMap) -- change the ring of an arrangement
- `substitute(Arrangement,Ring)` -- see substitute(Arrangement,RingMap) -- change the ring of an arrangement
- substitute(Arrangement,RingMap) -- change the ring of an arrangement
- trim(Arrangement) -- make a simple hyperplane arrangement
- `typeA(ZZ)` -- see typeA(ZZ,Ring) -- make the hyperplane arrangement defined by a type $A_n$ root system
- `typeA(ZZ,PolynomialRing)` -- see typeA(ZZ,Ring) -- make the hyperplane arrangement defined by a type $A_n$ root system
- typeA(ZZ,Ring) -- make the hyperplane arrangement defined by a type $A_n$ root system
- `typeB(ZZ)` -- see typeB(ZZ,Ring) -- make the hyperplane arrangement defined by a type $B_n$ root system
- `typeB(ZZ,PolynomialRing)` -- see typeB(ZZ,Ring) -- make the hyperplane arrangement defined by a type $B_n$ root system
- typeB(ZZ,Ring) -- make the hyperplane arrangement defined by a type $B_n$ root system
- `typeD(ZZ)` -- see typeD(ZZ,Ring) -- make the hyperplane arrangement defined by a type $D_n$ root system
- `typeD(ZZ,PolynomialRing)` -- see typeD(ZZ,Ring) -- make the hyperplane arrangement defined by a type $D_n$ root system
- typeD(ZZ,Ring) -- make the hyperplane arrangement defined by a type $D_n$ root system

## Methods that use a hyperplane arrangement:

- Arrangement == Arrangement -- whether two hyperplane arrangements are equal
- `closure(Arrangement,Ideal)` -- see closure(Arrangement,List) -- closure operation in the intersection lattice
- closure(Arrangement,List) -- closure operation in the intersection lattice
- coefficients(Arrangement) -- make a matrix from the coefficients of the defining equations
- cone(Arrangement,RingElement) -- creates an associated central hyperplane arrangement
- `cone(Arrangement,Symbol)` -- see cone(Arrangement,RingElement) -- creates an associated central hyperplane arrangement
- `EPY(Arrangement)` -- see EPY(Arrangement,PolynomialRing) -- compute the Eisenbud-Popescu-Yuzvinsky module of an arrangement
- EPY(Arrangement,PolynomialRing) -- compute the Eisenbud-Popescu-Yuzvinsky module of an arrangement
- flat(Arrangement,List) -- make a flat from a list of indices
- `flats(Arrangement)` -- see flats(ZZ,Arrangement) -- list the flats of an arrangement of a given rank
- flats(ZZ,Arrangement) -- list the flats of an arrangement of a given rank
- hyperplanes(Arrangement) -- the defining linear forms of an arrangement
- `toList(Arrangement)` -- see hyperplanes(Arrangement) -- the defining linear forms of an arrangement
- isCentral(Arrangement) -- test to see if a hyperplane arrangement is central
- matrix(Arrangement) -- make a matrix from the defining equations
- `orlikSolomon(Arrangement)` -- see orlikSolomon(Arrangement,PolynomialRing) -- compute the defining ideal for the Orlik-Solomon algebra
- orlikSolomon(Arrangement,PolynomialRing) -- compute the defining ideal for the Orlik-Solomon algebra
- `orlikSolomon(Arrangement,Ring)` -- see orlikSolomon(Arrangement,PolynomialRing) -- compute the defining ideal for the Orlik-Solomon algebra
- `orlikSolomon(Arrangement,Symbol)` -- see orlikSolomon(Arrangement,PolynomialRing) -- compute the defining ideal for the Orlik-Solomon algebra
- poincare(Arrangement) -- compute the Poincaré polynomial of an arrangement
- ring(Arrangement) -- get the underlying ring of a hyperplane arrangement

## For the programmer

The object Arrangement is a type, with ancestor classes HashTable < Thing.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/HyperplaneArrangements.m2:1087:0.