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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

TESTS:

sage: h = hyperplane_arrangements.Catalan(5)
sage: h.characteristic_polynomial()
x^5 - 30*x^4 + 335*x^3 - 1650*x^2 + 3024*x
sage: h.characteristic_polynomial.clear_cache()  # long time
sage: h.characteristic_polynomial()              # long time
x^5 - 30*x^4 + 335*x^3 - 1650*x^2 + 3024*x
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

TESTS:

sage: a.characteristic_polynomial.clear_cache()  # long time
sage: a.characteristic_polynomial()              # long time
x^3 - 6*x^2 + 9*x

REFERENCES:

[AR]D. Armstrong, B. Rhoades “The Shi arrangement and the Ish arrangement” Arxiv 1009.1655
Shi(n, K=Rational Field, names=None)

Return the Shi arrangement.

INPUT:

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

OUTPUT:

The Shi arrangement is the set of \(n(n-1)\) hyperplanes: \(\{ x_i - x_j = 0,1 : 1 \leq i \leq j \leq n \}\).

EXAMPLES:

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

TESTS:

sage: h = hyperplane_arrangements.Shi(4)
sage: h.characteristic_polynomial()
x^4 - 12*x^3 + 48*x^2 - 64*x
sage: h.characteristic_polynomial.clear_cache()  # long time
sage: h.characteristic_polynomial()              # long time
x^4 - 12*x^3 + 48*x^2 - 64*x
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:

[BigraphicalArrangements]S. Hopkins, D. Perkinson. “Bigraphical Arrangements”. Arxiv 1212.4398
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

TESTS:

sage: h = hyperplane_arrangements.graphical(g)
sage: h.characteristic_polynomial()
x^5 - 6*x^4 + 14*x^3 - 15*x^2 + 6*x
sage: h.characteristic_polynomial.clear_cache()  # long time
sage: h.characteristic_polynomial()              # long time
x^5 - 6*x^4 + 14*x^3 - 15*x^2 + 6*x
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

TESTS:

sage: h = hyperplane_arrangements.linial(5)
sage: h.characteristic_polynomial()
x^5 - 10*x^4 + 45*x^3 - 100*x^2 + 90*x
sage: h.characteristic_polynomial.clear_cache()  # long time
sage: h.characteristic_polynomial()              # long time
x^5 - 10*x^4 + 45*x^3 - 100*x^2 + 90*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

TESTS:

sage: h = hyperplane_arrangements.semiorder(5)
sage: h.characteristic_polynomial()
x^5 - 20*x^4 + 180*x^3 - 790*x^2 + 1380*x
sage: h.characteristic_polynomial.clear_cache()  # long time
sage: h.characteristic_polynomial()              # long time 
x^5 - 20*x^4 + 180*x^3 - 790*x^2 + 1380*x
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
  • dimenison – 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

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