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')[source]¶: Construct a parent class of Associahedra according to backend.

See also

Associahedra_base.

class sage.combinat.root_system.associahedron.Associahedra_base[source]¶

Bases: object

Base class of parent of Associahedra of specified dimension

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> from sage.combinat.root_system.associahedron import Associahedra
>>> parent = Associahedra(QQ,Integer(2),'ppl');  parent
Polyhedra in QQ^2
>>> type(parent)
<class 'sage.combinat.root_system.associahedron.Associahedra_ppl_with_category'>
>>> parent(['A',Integer(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

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

Python

>>> from sage.all import *
>>> parent(['A',Integer(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)[source]¶

Bases: Associahedra_base, Polyhedra_QQ_cdd

Element[source]¶: alias of Associahedron_class_cdd

class sage.combinat.root_system.associahedron.Associahedra_field(base_ring, ambient_dim, backend)[source]¶

Bases: Associahedra_base, Polyhedra_field

Element[source]¶: alias of Associahedron_class_field

class sage.combinat.root_system.associahedron.Associahedra_normaliz(base_ring, ambient_dim, backend)[source]¶

Bases: Associahedra_base, Polyhedra_QQ_normaliz

Element[source]¶: alias of Associahedron_class_normaliz

class sage.combinat.root_system.associahedron.Associahedra_polymake(base_ring, ambient_dim, backend)[source]¶

Bases: Associahedra_base, Polyhedra_polymake

Element[source]¶: alias of Associahedron_class_polymake

class sage.combinat.root_system.associahedron.Associahedra_ppl(base_ring, ambient_dim, backend)[source]¶

Bases: Associahedra_base, Polyhedra_QQ_ppl

Element[source]¶: alias of Associahedron_class_ppl

sage.combinat.root_system.associahedron.Associahedron(cartan_type, backend='ppl')[source]¶

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:

cartan_type – a cartan type according to sage.combinat.root_system.cartan_type.CartanTypeFactory
backend – string ('ppl'); the backend to use; see sage.geometry.polyhedron.constructor.Polyhedron()

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> Asso = polytopes.associahedron(['A',Integer(2)]); Asso
Generalized associahedron of type ['A', 2] with 5 vertices

>>> 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]

>>> 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))

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

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

Sage

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

Python

>>> from sage.all import *
>>> Asso = polytopes.associahedron(['A',Integer(3)]); Asso
Generalized associahedron of type ['A', 3] with 14 vertices
>>> Asso.plot()  # long time
Graphics3d Object

>>> Asso = polytopes.associahedron(['B',Integer(3)]); Asso
Generalized associahedron of type ['B', 3] with 20 vertices
>>> Asso.plot()  # long time
Graphics3d Object

class sage.combinat.root_system.associahedron.Associahedron_class_base(parent=None, Vrep=None, Hrep=None, cartan_type=None, **kwds)[source]¶

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()[source]¶

Return the Cartan type of self.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> polytopes.associahedron(['A',Integer(3)]).cartan_type()
['A', 3]

vertices_in_root_space()[source]¶

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

EXAMPLES:

Sage

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])

Python

>>> from sage.all import *
>>> Asso = polytopes.associahedron(['A',Integer(2)])
>>> 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))

>>> 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=None, Vrep=None, Hrep=None, cartan_type=None, **kwds)[source]¶: Bases: Associahedron_class_base, Polyhedron_QQ_cdd

class sage.combinat.root_system.associahedron.Associahedron_class_field(parent=None, Vrep=None, Hrep=None, cartan_type=None, **kwds)[source]¶: Bases: Associahedron_class_base, Polyhedron_field

class sage.combinat.root_system.associahedron.Associahedron_class_normaliz(parent=None, Vrep=None, Hrep=None, cartan_type=None, **kwds)[source]¶: Bases: Associahedron_class_base, Polyhedron_QQ_normaliz

class sage.combinat.root_system.associahedron.Associahedron_class_polymake(parent=None, Vrep=None, Hrep=None, cartan_type=None, **kwds)[source]¶: Bases: Associahedron_class_base, Polyhedron_polymake

class sage.combinat.root_system.associahedron.Associahedron_class_ppl(parent=None, Vrep=None, Hrep=None, cartan_type=None, **kwds)[source]¶: Bases: Associahedron_class_base, Polyhedron_QQ_ppl