Associahedron

Todo

• fix adjacency matrix
• edit graph method to get proper vertex labellings
• UniqueRepresentation?

AUTHORS:

• Christian Stump

class sage.combinat.root_system.associahedron.Associahedra(base_ring, ambient_dim)

Bases: sage.geometry.polyhedron.parent.Polyhedra_QQ_ppl

Parent of Associahedra of specified dimension

EXAMPLES:

sage: from sage.combinat.root_system.associahedron import Associahedra
sage: parent = Associahedra(QQ,2);  parent
Polyhedra in QQ^2
sage: type(parent)
<class 'sage.combinat.root_system.associahedron.Associahedra_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

Element

alias of Associahedron_class

sage.combinat.root_system.associahedron.Associahedron(cartan_type)

Construct an associahedron

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 Wikipedia article Associahedron

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

EXAMPLES:

sage: Asso = 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: Associahedron(['B',2])
Generalized associahedron of type ['B', 2] with 6 vertices

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

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

    sage: Asso = Associahedron(['B',3]); Asso
    Generalized associahedron of type ['B', 3] with 20 vertices
    sage: Asso.plot()
    Graphics3d Object

TESTS::

    sage: sorted(Associahedron(['A',3]).vertices())
    [A vertex at (-3/2, 0, -1/2), A vertex at (-3/2, 0, 3/2),
     A vertex at (-3/2, 1, -3/2), A vertex at (-3/2, 2, -3/2),
     A vertex at (-3/2, 2, 3/2), A vertex at (-1/2, -1, -1/2),
     A vertex at (-1/2, 0, -3/2), A vertex at (1/2, -2, 1/2),
     A vertex at (1/2, -2, 3/2), A vertex at (3/2, -2, 1/2),
     A vertex at (3/2, -2, 3/2), A vertex at (3/2, 0, -3/2),
     A vertex at (3/2, 2, -3/2), A vertex at (3/2, 2, 3/2)]

    sage: sorted(Associahedron(['B',3]).vertices())
    [A vertex at (-3, 0, 0), A vertex at (-3, 0, 3),
     A vertex at (-3, 2, -2), A vertex at (-3, 4, -3),
     A vertex at (-3, 5, -3), A vertex at (-3, 5, 3),
     A vertex at (-2, 1, -2), A vertex at (-2, 3, -3),
     A vertex at (-1, -2, 0), A vertex at (-1, -1, -1),
     A vertex at (1, -4, 1), A vertex at (1, -3, 0),
     A vertex at (2, -5, 2), A vertex at (2, -5, 3),
     A vertex at (3, -5, 2), A vertex at (3, -5, 3),
     A vertex at (3, -3, 0), A vertex at (3, 3, -3),
     A vertex at (3, 5, -3), A vertex at (3, 5, 3)]

    sage: Associahedron(['A',4]).f_vector()
    (1, 42, 84, 56, 14, 1)
    sage: Associahedron(['B',4]).f_vector()
    (1, 70, 140, 90, 20, 1)

REFERENCES:

- [CFZ] Chapoton, Fomin, Zelevinsky - Polytopal realizations of
  generalized associahedra, arXiv:0202004.

class sage.combinat.root_system.associahedron.Associahedron_class(parent, Vrep, Hrep, **kwds)

Bases: sage.geometry.polyhedron.backend_ppl.Polyhedron_QQ_ppl

The Python class of an associahedron

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

TESTS:

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

cartan_type()

Returns the Cartan type

EXAMPLES:

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

vertices_in_root_space()

Returns the vertices of self as elements in the root space

EXAMPLES:

sage: Asso = 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])