A collection of useful or interesting hyperplane arrangements. See sage.geometry.hyperplane_arrangement.arrangement for details about how to construct your own hyperplane arrangements.
Bases: object
The library of hyperplane arrangements.
Return the Catalan arrangement.
INPUT:
OUTPUT:
The arrangement of hyperplanes
.
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
Return the Shi hyperplane arrangement of a graph .
INPUT:
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
Return the semiorder hyperplane arrangement of a graph.
INPUT:
OUTPUT:
The semiorder hyperplane arrangement of a graph G is the
arrangement where
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
Return the Ish arrangement.
INPUT:
OUTPUT:
The Ish arrangement, which is the set of hyperplanes.
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 |
Return the Shi arrangement.
INPUT:
OUTPUT:
The Shi arrangement is the set of hyperplanes:
.
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
Return a bigraphical hyperplane arrangement.
INPUT:
OUTPUT:
The hyperplane arrangement with hyperplanes and
for each edge
of
G. The indices
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 |
The braid arrangement.
INPUT:
OUTPUT:
The hyperplane arrangement consisting of the
hyperplanes
.
EXAMPLES:
sage: hyperplane_arrangements.braid(4)
Arrangement of 6 hyperplanes of dimension 4 and rank 3
Return the coordinate hyperplane arrangement.
INPUT:
OUTPUT:
The coordinate hyperplane arrangement, which is the central
hyperplane arrangement consisting of the coordinate
hyperplanes .
EXAMPLES:
sage: hyperplane_arrangements.coordinate(5)
Arrangement of 5 hyperplanes of dimension 5 and rank 5
Return the graphical hyperplane arrangement of a graph G.
INPUT:
OUTPUT:
The graphical hyperplane arrangement of a graph G, which is
the arrangement for all edges
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
Return the linial hyperplane arrangement.
INPUT:
OUTPUT:
The linial hyperplane arrangement is the set of hyperplanes
.
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
Return the semiorder arrangement.
INPUT:
OUTPUT:
The semiorder arrangement, which is the set of
hyperplanes
.
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
Construct the parent for the hyperplane arrangements.
For internal use only.
INPUT:
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