Associahedron

Todo

  • fix adjacency matrix

  • edit graph method to get proper vertex labellings

  • UniqueRepresentation?

AUTHORS:

  • Christian Stump

sage.combinat.root_system.associahedron.Associahedra(base_ring, ambient_dim, backend='ppl')

Construct a parent class of Associahedra according to backend.

See also

Associahedra_base.

class sage.combinat.root_system.associahedron.Associahedra_base

Bases: object

Base class of parent of Associahedra of specified dimension

EXAMPLES:

sage: from sage.combinat.root_system.associahedron import Associahedra
sage: parent = Associahedra(QQ,2,'ppl');  parent
Polyhedra in QQ^2
sage: type(parent)
<class 'sage.combinat.root_system.associahedron.Associahedra_ppl_with_category'>
sage: parent(['A',2])
Generalized associahedron of type ['A', 2] with 5 vertices

Importantly, the parent knows the dimension of the ambient space. If you try to construct an associahedron of a different dimension, a ValueError is raised:

sage: parent(['A',3])
Traceback (most recent call last):
...
ValueError: V-representation data requires a list of length ambient_dim
class sage.combinat.root_system.associahedron.Associahedra_cdd(base_ring, ambient_dim, backend)

Bases: sage.combinat.root_system.associahedron.Associahedra_base, sage.geometry.polyhedron.parent.Polyhedra_QQ_cdd

Element

alias of Associahedron_class_cdd

class sage.combinat.root_system.associahedron.Associahedra_field(base_ring, ambient_dim, backend)

Bases: sage.combinat.root_system.associahedron.Associahedra_base, sage.geometry.polyhedron.parent.Polyhedra_field

Element

alias of Associahedron_class_field

class sage.combinat.root_system.associahedron.Associahedra_normaliz(base_ring, ambient_dim, backend)

Bases: sage.combinat.root_system.associahedron.Associahedra_base, sage.geometry.polyhedron.parent.Polyhedra_QQ_normaliz

Element

alias of Associahedron_class_normaliz

class sage.combinat.root_system.associahedron.Associahedra_polymake(base_ring, ambient_dim, backend)

Bases: sage.combinat.root_system.associahedron.Associahedra_base, sage.geometry.polyhedron.parent.Polyhedra_polymake

Element

alias of Associahedron_class_polymake

class sage.combinat.root_system.associahedron.Associahedra_ppl(base_ring, ambient_dim, backend)

Bases: sage.combinat.root_system.associahedron.Associahedra_base, sage.geometry.polyhedron.parent.Polyhedra_QQ_ppl

Element

alias of Associahedron_class_ppl

sage.combinat.root_system.associahedron.Associahedron(cartan_type, backend='ppl')

Construct an associahedron.

The generalized associahedron is a polytopal complex with vertices in one-to-one correspondence with clusters in the cluster complex, and with edges between two vertices if and only if the associated two clusters intersect in codimension 1.

The associahedron of type \(A_n\) is one way to realize the classical associahedron as defined in the Wikipedia article Associahedron.

A polytopal realization of the associahedron can be found in [CFZ2002]. The implementation is based on [CFZ2002], Theorem 1.5, Remark 1.6, and Corollary 1.9.

INPUT:

EXAMPLES:

sage: Asso = polytopes.associahedron(['A',2]); Asso
Generalized associahedron of type ['A', 2] with 5 vertices

sage: sorted(Asso.Hrepresentation(), key=repr)
[An inequality (-1, 0) x + 1 >= 0,
 An inequality (0, -1) x + 1 >= 0,
 An inequality (0, 1) x + 1 >= 0,
 An inequality (1, 0) x + 1 >= 0,
 An inequality (1, 1) x + 1 >= 0]

sage: Asso.Vrepresentation()
(A vertex at (1, -1), A vertex at (1, 1), A vertex at (-1, 1),
 A vertex at (-1, 0), A vertex at (0, -1))

sage: polytopes.associahedron(['B',2])
Generalized associahedron of type ['B', 2] with 6 vertices

The two pictures of [CFZ2002] can be recovered with:

sage: Asso = polytopes.associahedron(['A',3]); Asso
Generalized associahedron of type ['A', 3] with 14 vertices
sage: Asso.plot()  # long time
Graphics3d Object

sage: Asso = polytopes.associahedron(['B',3]); Asso
Generalized associahedron of type ['B', 3] with 20 vertices
sage: Asso.plot()  # long time
Graphics3d Object
class sage.combinat.root_system.associahedron.Associahedron_class_base(parent, Vrep, Hrep, cartan_type=None, **kwds)

Bases: object

The base class of the Python class of an associahedron

You should use the Associahedron() convenience function to construct associahedra from the Cartan type.

cartan_type()

Return the Cartan type of self.

EXAMPLES:

sage: polytopes.associahedron(['A',3]).cartan_type()
['A', 3]
vertices_in_root_space()

Return the vertices of self as elements in the root space.

EXAMPLES:

sage: Asso = polytopes.associahedron(['A',2])
sage: Asso.vertices()
(A vertex at (1, -1), A vertex at (1, 1),
 A vertex at (-1, 1), A vertex at (-1, 0),
 A vertex at (0, -1))

sage: Asso.vertices_in_root_space()
(alpha[1] - alpha[2], alpha[1] + alpha[2], -alpha[1] + alpha[2],
-alpha[1], -alpha[2])
class sage.combinat.root_system.associahedron.Associahedron_class_cdd(parent, Vrep, Hrep, cartan_type=None, **kwds)

Bases: sage.combinat.root_system.associahedron.Associahedron_class_base, sage.geometry.polyhedron.backend_cdd.Polyhedron_QQ_cdd

class sage.combinat.root_system.associahedron.Associahedron_class_field(parent, Vrep, Hrep, cartan_type=None, **kwds)

Bases: sage.combinat.root_system.associahedron.Associahedron_class_base, sage.geometry.polyhedron.backend_field.Polyhedron_field

class sage.combinat.root_system.associahedron.Associahedron_class_normaliz(parent, Vrep, Hrep, cartan_type=None, **kwds)

Bases: sage.combinat.root_system.associahedron.Associahedron_class_base, sage.geometry.polyhedron.backend_normaliz.Polyhedron_QQ_normaliz

class sage.combinat.root_system.associahedron.Associahedron_class_polymake(parent, Vrep, Hrep, cartan_type=None, **kwds)

Bases: sage.combinat.root_system.associahedron.Associahedron_class_base, sage.geometry.polyhedron.backend_polymake.Polyhedron_polymake

class sage.combinat.root_system.associahedron.Associahedron_class_ppl(parent, Vrep, Hrep, cartan_type=None, **kwds)

Bases: sage.combinat.root_system.associahedron.Associahedron_class_base, sage.geometry.polyhedron.backend_ppl.Polyhedron_QQ_ppl