Library of Hyperplane Arrangements¶

A collection of useful or interesting hyperplane arrangements. See sage.geometry.hyperplane_arrangement.arrangement for details about how to construct your own hyperplane arrangements.

class sage.geometry.hyperplane_arrangement.library.HyperplaneArrangementLibrary¶

Bases: object

The library of hyperplane arrangements.

Catalan(n, K=Rational Field, names=None)¶

Return the Catalan arrangement.

INPUT:

n – integer
K – field (default: $\QQ$ )
names – tuple of strings or None (default); the variable names for the ambient space

OUTPUT:

The arrangement of $3n(n-1)/2$ hyperplanes $\{ x_i - x_j = -1,0,1 : 1 \leq i \leq j \leq n \}$ .

EXAMPLES:

sage: hyperplane_arrangements.Catalan(5)
Arrangement of 30 hyperplanes of dimension 5 and rank 4

G_Shi(G, K=Rational Field, names=None)¶

Return the Shi hyperplane arrangement of a graph $G$ .

INPUT:

G – graph
K – field (default: $\QQ$ )
names – tuple of strings or None (default); the variable names for the ambient space

OUTPUT:

The Shi hyperplane arrangement of the given graph G.

EXAMPLES:

sage: G = graphs.CompleteGraph(5)
sage: hyperplane_arrangements.G_Shi(G)
Arrangement of 20 hyperplanes of dimension 5 and rank 4
sage: g = graphs.HouseGraph()
sage: hyperplane_arrangements.G_Shi(g)
Arrangement of 12 hyperplanes of dimension 5 and rank 4
sage: a = hyperplane_arrangements.G_Shi(graphs.WheelGraph(4));  a
Arrangement of 12 hyperplanes of dimension 4 and rank 3

G_semiorder(G, K=Rational Field, names=None)¶

Return the semiorder hyperplane arrangement of a graph.

INPUT:

G – graph
K – field (default: $\QQ$ )
names – tuple of strings or None (default); the variable names for the ambient space

OUTPUT:

The semiorder hyperplane arrangement of a graph G is the arrangement $\{ x_i - x_j = -1,1 \}$ where $ij$ is an edge of G.

EXAMPLES:

sage: G = graphs.CompleteGraph(5)
sage: hyperplane_arrangements.G_semiorder(G)
Arrangement of 20 hyperplanes of dimension 5 and rank 4
sage: g = graphs.HouseGraph()
sage: hyperplane_arrangements.G_semiorder(g)
Arrangement of 12 hyperplanes of dimension 5 and rank 4

Ish(n, K=Rational Field, names=None)¶

Return the Ish arrangement.

INPUT:

n – integer
K – field (default:QQ)
names – tuple of strings or None (default); the variable names for the ambient space

OUTPUT:

The Ish arrangement, which is the set of $n(n-1)$ hyperplanes.

$\{ x_i - x_j = 0 : 1 \leq i \leq j \leq n \} \cup \{ x_1 - x_j = i : 1 \leq i \leq j \leq n \}.$

EXAMPLES:

sage: a = hyperplane_arrangements.Ish(3);  a
Arrangement of 6 hyperplanes of dimension 3 and rank 2
sage: a.characteristic_polynomial()
x^3 - 6*x^2 + 9*x
sage: b = hyperplane_arrangements.Shi(3)
sage: b.characteristic_polynomial()
x^3 - 6*x^2 + 9*x

REFERENCES:

[AR2012]

Shi(data, K=Rational Field, names=None, m=1)¶

Return the Shi arrangement.

INPUT:

data – either an integer or a Cartan type (or coercible into; see “CartanType”)
K – field (default:QQ)
names – tuple of strings or None (default); the variable names for the ambient space
m – integer (default: 1)

OUTPUT:

If data is an integer $n$ , return the Shi arrangement in dimension $n$ , i.e. the set of $n(n-1)$ hyperplanes: $\{ x_i - x_j = 0,1 : 1 \leq i \leq j \leq n \}$ . This corresponds to the Shi arrangement of Cartan type $A_{n-1}$ .
If data is a Cartan type, return the Shi arrangement of given type.
If $m > 1$ , return the $m$ -extended Shi arrangement of given type.

The $m$ -extended Shi arrangement of a given crystallographic Cartan type is defined by the inner product $\langle a,x \rangle = k$ for $-m < k \leq m$ and $a \in \Phi^+$ is a positive root of the root system $\Phi$ .

EXAMPLES:

sage: hyperplane_arrangements.Shi(4)
Arrangement of 12 hyperplanes of dimension 4 and rank 3
sage: hyperplane_arrangements.Shi("A3")
Arrangement of 12 hyperplanes of dimension 4 and rank 3
sage: hyperplane_arrangements.Shi("A3",m=2)
Arrangement of 24 hyperplanes of dimension 4 and rank 3
sage: hyperplane_arrangements.Shi("B4")
Arrangement of 32 hyperplanes of dimension 4 and rank 4
sage: hyperplane_arrangements.Shi("B4",m=3)
Arrangement of 96 hyperplanes of dimension 4 and rank 4
sage: hyperplane_arrangements.Shi("C3")
Arrangement of 18 hyperplanes of dimension 3 and rank 3
sage: hyperplane_arrangements.Shi("D4",m=3)
Arrangement of 72 hyperplanes of dimension 4 and rank 4
sage: hyperplane_arrangements.Shi("E6")
Arrangement of 72 hyperplanes of dimension 8 and rank 6
sage: hyperplane_arrangements.Shi("E6",m=2)
Arrangement of 144 hyperplanes of dimension 8 and rank 6

If the Cartan type is not crystallographic, the Shi arrangement is not defined:

sage: hyperplane_arrangements.Shi("H4")
Traceback (most recent call last):
...
NotImplementedError: Shi arrangements are not defined for non crystallographic Cartan types

The characteristic polynomial is pre-computed using the results of [Ath1996]:

sage: hyperplane_arrangements.Shi("A3").characteristic_polynomial()
x^4 - 12*x^3 + 48*x^2 - 64*x
sage: hyperplane_arrangements.Shi("A3",m=2).characteristic_polynomial()
x^4 - 24*x^3 + 192*x^2 - 512*x
sage: hyperplane_arrangements.Shi("C3").characteristic_polynomial()
x^3 - 18*x^2 + 108*x - 216
sage: hyperplane_arrangements.Shi("E6").characteristic_polynomial()
x^8 - 72*x^7 + 2160*x^6 - 34560*x^5 + 311040*x^4 - 1492992*x^3 + 2985984*x^2
sage: hyperplane_arrangements.Shi("B4",m=3).characteristic_polynomial()
x^4 - 96*x^3 + 3456*x^2 - 55296*x + 331776

bigraphical(G, A=None, K=Rational Field, names=None)¶

Return a bigraphical hyperplane arrangement.

INPUT:

G – graph
A – list, matrix, dictionary (default: None gives semiorder), or the string ‘generic’
K – field (default: $\QQ$ )
names – tuple of strings or None (default); the variable names for the ambient space

OUTPUT:

The hyperplane arrangement with hyperplanes $x_i - x_j = A[i,j]$ and $x_j - x_i = A[j,i]$ for each edge $v_i, v_j$ of G. The indices $i,j$ are the indices of elements of G.vertices().

EXAMPLES:

sage: G = graphs.CycleGraph(4)
sage: G.edges()
[(0, 1, None), (0, 3, None), (1, 2, None), (2, 3, None)]
sage: G.edges(labels=False)
[(0, 1), (0, 3), (1, 2), (2, 3)]
sage: A = {0:{1:1, 3:2}, 1:{0:3, 2:0}, 2:{1:2, 3:1}, 3:{2:0, 0:2}}
sage: HA = hyperplane_arrangements.bigraphical(G, A)
sage: HA.n_regions()
63
sage: hyperplane_arrangements.bigraphical(G, 'generic').n_regions()
65
sage: hyperplane_arrangements.bigraphical(G).n_regions()
59

REFERENCES:

[HP2016]

braid(n, K=Rational Field, names=None)¶

The braid arrangement.

INPUT:

n – integer
K – field (default: QQ)
names – tuple of strings or None (default); the variable names for the ambient space

OUTPUT:

The hyperplane arrangement consisting of the $n(n-1)/2$ hyperplanes $\{ x_i - x_j = 0 : 1 \leq i \leq j \leq n \}$ .

EXAMPLES:

sage: hyperplane_arrangements.braid(4)
Arrangement of 6 hyperplanes of dimension 4 and rank 3

coordinate(n, K=Rational Field, names=None)¶

Return the coordinate hyperplane arrangement.

INPUT:

n – integer
K – field (default: $\QQ$ )
names – tuple of strings or None (default); the variable names for the ambient space

OUTPUT:

The coordinate hyperplane arrangement, which is the central hyperplane arrangement consisting of the coordinate hyperplanes $x_i = 0$ .

EXAMPLES:

sage: hyperplane_arrangements.coordinate(5)
Arrangement of 5 hyperplanes of dimension 5 and rank 5

graphical(G, K=Rational Field, names=None)¶

Return the graphical hyperplane arrangement of a graph G.

INPUT:

G – graph
K – field (default: $\QQ$ )
names – tuple of strings or None (default); the variable names for the ambient space

OUTPUT:

The graphical hyperplane arrangement of a graph G, which is the arrangement $\{ x_i - x_j = 0 \}$ for all edges $ij$ of the graph G.

EXAMPLES:

sage: G = graphs.CompleteGraph(5)
sage: hyperplane_arrangements.graphical(G)
Arrangement of 10 hyperplanes of dimension 5 and rank 4
sage: g = graphs.HouseGraph()
sage: hyperplane_arrangements.graphical(g)
Arrangement of 6 hyperplanes of dimension 5 and rank 4

linial(n, K=Rational Field, names=None)¶

Return the linial hyperplane arrangement.

INPUT:

n – integer
K – field (default: $\QQ$ )
names – tuple of strings or None (default); the variable names for the ambient space

OUTPUT:

The linial hyperplane arrangement is the set of hyperplanes $\{x_i - x_j = 1 : 1\leq i < j \leq n\}$ .

EXAMPLES:

sage: a = hyperplane_arrangements.linial(4);  a
Arrangement of 6 hyperplanes of dimension 4 and rank 3
sage: a.characteristic_polynomial()
x^4 - 6*x^3 + 15*x^2 - 14*x

semiorder(n, K=Rational Field, names=None)¶

Return the semiorder arrangement.

INPUT:

n – integer
K – field (default: $\QQ$ )
names – tuple of strings or None (default); the variable names for the ambient space

OUTPUT:

The semiorder arrangement, which is the set of $n(n-1)$ hyperplanes $\{ x_i - x_j = -1,1 : 1 \leq i \leq j \leq n\}$ .

EXAMPLES:

sage: hyperplane_arrangements.semiorder(4)
Arrangement of 12 hyperplanes of dimension 4 and rank 3

sage.geometry.hyperplane_arrangement.library.make_parent(base_ring, dimension, names=None)¶

Construct the parent for the hyperplane arrangements.

For internal use only.

INPUT:

base_ring – a ring
dimension – integer
names – None (default) or a list/tuple/iterable of strings

OUTPUT:

A new HyperplaneArrangements instance.

EXAMPLES:

sage: from sage.geometry.hyperplane_arrangement.library import make_parent
sage: make_parent(QQ, 3)
Hyperplane arrangements in 3-dimensional linear space over
Rational Field with coordinates t0, t1, t2

Library of Hyperplane Arrangements¶

Previous topic

Next topic

This Page