Coxeter Types

class sage.combinat.root_system.coxeter_type.CoxeterType[source]

Bases: SageObject

Abstract class for Coxeter types.

bilinear_form(R=None)[source]

Return the bilinear form over R associated to self.

INPUT:

  • R – (default: universal cyclotomic field) a ring used to compute the bilinear form

EXAMPLES:

sage: # needs sage.graphs sage.libs.gap
sage: CoxeterType(['A', 2, 1]).bilinear_form()
[   1 -1/2 -1/2]
[-1/2    1 -1/2]
[-1/2 -1/2    1]
sage: CoxeterType(['H', 3]).bilinear_form()
[                      1                    -1/2                       0]
[                   -1/2                       1 1/2*E(5)^2 + 1/2*E(5)^3]
[                      0 1/2*E(5)^2 + 1/2*E(5)^3                       1]
sage: C = CoxeterMatrix([[1,-1,-1],[-1,1,-1],[-1,-1,1]])
sage: C.bilinear_form()
[ 1 -1 -1]
[-1  1 -1]
[-1 -1  1]

>>> from sage.all import *
>>> # needs sage.graphs sage.libs.gap
>>> CoxeterType(['A', Integer(2), Integer(1)]).bilinear_form()
[   1 -1/2 -1/2]
[-1/2    1 -1/2]
[-1/2 -1/2    1]
>>> CoxeterType(['H', Integer(3)]).bilinear_form()
[                      1                    -1/2                       0]
[                   -1/2                       1 1/2*E(5)^2 + 1/2*E(5)^3]
[                      0 1/2*E(5)^2 + 1/2*E(5)^3                       1]
>>> C = CoxeterMatrix([[Integer(1),-Integer(1),-Integer(1)],[-Integer(1),Integer(1),-Integer(1)],[-Integer(1),-Integer(1),Integer(1)]])
>>> C.bilinear_form()
[ 1 -1 -1]
[-1  1 -1]
[-1 -1  1]
coxeter_graph()[source]

Return the Coxeter graph associated to self.

EXAMPLES:

sage: CoxeterType(['A', 3]).coxeter_graph()                                 # needs sage.graphs
Graph on 3 vertices
sage: CoxeterType(['A', 3, 1]).coxeter_graph()                              # needs sage.graphs
Graph on 4 vertices

>>> from sage.all import *
>>> CoxeterType(['A', Integer(3)]).coxeter_graph()                                 # needs sage.graphs
Graph on 3 vertices
>>> CoxeterType(['A', Integer(3), Integer(1)]).coxeter_graph()                              # needs sage.graphs
Graph on 4 vertices
coxeter_matrix()[source]

Return the Coxeter matrix associated to self.

EXAMPLES:

sage: CoxeterType(['A', 3]).coxeter_matrix()                                # needs sage.graphs
[1 3 2]
[3 1 3]
[2 3 1]
sage: CoxeterType(['A', 3, 1]).coxeter_matrix()                             # needs sage.graphs
[1 3 2 3]
[3 1 3 2]
[2 3 1 3]
[3 2 3 1]

>>> from sage.all import *
>>> CoxeterType(['A', Integer(3)]).coxeter_matrix()                                # needs sage.graphs
[1 3 2]
[3 1 3]
[2 3 1]
>>> CoxeterType(['A', Integer(3), Integer(1)]).coxeter_matrix()                             # needs sage.graphs
[1 3 2 3]
[3 1 3 2]
[2 3 1 3]
[3 2 3 1]
index_set()[source]

Return the index set for self.

This is the list of the nodes of the associated Coxeter graph.

EXAMPLES:

sage: CoxeterType(['A', 3, 1]).index_set()
(0, 1, 2, 3)
sage: CoxeterType(['D', 4]).index_set()
(1, 2, 3, 4)
sage: CoxeterType(['A', 7, 2]).index_set()
(0, 1, 2, 3, 4)
sage: CoxeterType(['A', 7, 2]).index_set()
(0, 1, 2, 3, 4)
sage: CoxeterType(['A', 6, 2]).index_set()
(0, 1, 2, 3)
sage: CoxeterType(['D', 6, 2]).index_set()
(0, 1, 2, 3, 4, 5)
sage: CoxeterType(['E', 6, 1]).index_set()
(0, 1, 2, 3, 4, 5, 6)
sage: CoxeterType(['E', 6, 2]).index_set()
(0, 1, 2, 3, 4)
sage: CoxeterType(['A', 2, 2]).index_set()
(0, 1)
sage: CoxeterType(['G', 2, 1]).index_set()
(0, 1, 2)
sage: CoxeterType(['F', 4, 1]).index_set()
(0, 1, 2, 3, 4)

>>> from sage.all import *
>>> CoxeterType(['A', Integer(3), Integer(1)]).index_set()
(0, 1, 2, 3)
>>> CoxeterType(['D', Integer(4)]).index_set()
(1, 2, 3, 4)
>>> CoxeterType(['A', Integer(7), Integer(2)]).index_set()
(0, 1, 2, 3, 4)
>>> CoxeterType(['A', Integer(7), Integer(2)]).index_set()
(0, 1, 2, 3, 4)
>>> CoxeterType(['A', Integer(6), Integer(2)]).index_set()
(0, 1, 2, 3)
>>> CoxeterType(['D', Integer(6), Integer(2)]).index_set()
(0, 1, 2, 3, 4, 5)
>>> CoxeterType(['E', Integer(6), Integer(1)]).index_set()
(0, 1, 2, 3, 4, 5, 6)
>>> CoxeterType(['E', Integer(6), Integer(2)]).index_set()
(0, 1, 2, 3, 4)
>>> CoxeterType(['A', Integer(2), Integer(2)]).index_set()
(0, 1)
>>> CoxeterType(['G', Integer(2), Integer(1)]).index_set()
(0, 1, 2)
>>> CoxeterType(['F', Integer(4), Integer(1)]).index_set()
(0, 1, 2, 3, 4)
is_affine()[source]

Return whether self is affine.

EXAMPLES:

sage: CoxeterType(['A', 3]).is_affine()
False
sage: CoxeterType(['A', 3, 1]).is_affine()
True

>>> from sage.all import *
>>> CoxeterType(['A', Integer(3)]).is_affine()
False
>>> CoxeterType(['A', Integer(3), Integer(1)]).is_affine()
True
is_crystallographic()[source]

Return whether self is crystallographic.

This returns False by default. Derived class should override this appropriately.

EXAMPLES:

sage: [ [t, t.is_crystallographic() ] for t in CartanType.samples(finite=True) ]
[[['A', 1], True], [['A', 5], True],
 [['B', 1], True], [['B', 5], True],
 [['C', 1], True], [['C', 5], True],
 [['D', 2], True], [['D', 3], True], [['D', 5], True],
 [['E', 6], True], [['E', 7], True], [['E', 8], True],
 [['F', 4], True], [['G', 2], True],
 [['I', 5], False], [['H', 3], False], [['H', 4], False]]

>>> from sage.all import *
>>> [ [t, t.is_crystallographic() ] for t in CartanType.samples(finite=True) ]
[[['A', 1], True], [['A', 5], True],
 [['B', 1], True], [['B', 5], True],
 [['C', 1], True], [['C', 5], True],
 [['D', 2], True], [['D', 3], True], [['D', 5], True],
 [['E', 6], True], [['E', 7], True], [['E', 8], True],
 [['F', 4], True], [['G', 2], True],
 [['I', 5], False], [['H', 3], False], [['H', 4], False]]
is_finite()[source]

Return whether self is finite.

EXAMPLES:

sage: CoxeterType(['A',4]).is_finite()
True
sage: CoxeterType(['A',4, 1]).is_finite()
False

>>> from sage.all import *
>>> CoxeterType(['A',Integer(4)]).is_finite()
True
>>> CoxeterType(['A',Integer(4), Integer(1)]).is_finite()
False
is_simply_laced()[source]

Return whether self is simply laced.

This returns False by default. Derived class should override this appropriately.

EXAMPLES:

sage: [ [t, t.is_simply_laced() ] for t in CartanType.samples() ]
[[['A', 1], True], [['A', 5], True],
 [['B', 1], True], [['B', 5], False],
 [['C', 1], True], [['C', 5], False],
 [['D', 2], True], [['D', 3], True], [['D', 5], True],
 [['E', 6], True], [['E', 7], True], [['E', 8], True],
 [['F', 4], False], [['G', 2], False],
 [['I', 5], False], [['H', 3], False], [['H', 4], False],
 [['A', 1, 1], False], [['A', 5, 1], True],
 [['B', 1, 1], False], [['B', 5, 1], False],
 [['C', 1, 1], False], [['C', 5, 1], False],
 [['D', 3, 1], True], [['D', 5, 1], True],
 [['E', 6, 1], True], [['E', 7, 1], True], [['E', 8, 1], True],
 [['F', 4, 1], False], [['G', 2, 1], False],
 [['BC', 1, 2], False], [['BC', 5, 2], False],
 [['B', 5, 1]^*, False], [['C', 4, 1]^*, False],
 [['F', 4, 1]^*, False], [['G', 2, 1]^*, False],
 [['BC', 1, 2]^*, False], [['BC', 5, 2]^*, False]]

>>> from sage.all import *
>>> [ [t, t.is_simply_laced() ] for t in CartanType.samples() ]
[[['A', 1], True], [['A', 5], True],
 [['B', 1], True], [['B', 5], False],
 [['C', 1], True], [['C', 5], False],
 [['D', 2], True], [['D', 3], True], [['D', 5], True],
 [['E', 6], True], [['E', 7], True], [['E', 8], True],
 [['F', 4], False], [['G', 2], False],
 [['I', 5], False], [['H', 3], False], [['H', 4], False],
 [['A', 1, 1], False], [['A', 5, 1], True],
 [['B', 1, 1], False], [['B', 5, 1], False],
 [['C', 1, 1], False], [['C', 5, 1], False],
 [['D', 3, 1], True], [['D', 5, 1], True],
 [['E', 6, 1], True], [['E', 7, 1], True], [['E', 8, 1], True],
 [['F', 4, 1], False], [['G', 2, 1], False],
 [['BC', 1, 2], False], [['BC', 5, 2], False],
 [['B', 5, 1]^*, False], [['C', 4, 1]^*, False],
 [['F', 4, 1]^*, False], [['G', 2, 1]^*, False],
 [['BC', 1, 2]^*, False], [['BC', 5, 2]^*, False]]
rank()[source]

Return the rank of self.

This is the number of nodes of the associated Coxeter graph.

EXAMPLES:

sage: CoxeterType(['A', 4]).rank()
4
sage: CoxeterType(['A', 7, 2]).rank()
5
sage: CoxeterType(['I', 8]).rank()
2

>>> from sage.all import *
>>> CoxeterType(['A', Integer(4)]).rank()
4
>>> CoxeterType(['A', Integer(7), Integer(2)]).rank()
5
>>> CoxeterType(['I', Integer(8)]).rank()
2
classmethod samples(finite=None, affine=None, crystallographic=None)[source]

Return a sample of the available Coxeter types.

INPUT:

  • finite – boolean or None (default: None)

  • affine – boolean or None (default: None)

  • crystallographic – boolean or None (default: None)

The sample contains all the exceptional finite and affine Coxeter types, as well as typical representatives of the infinite families.

EXAMPLES:

sage: CoxeterType.samples()
[Coxeter type of ['A', 1], Coxeter type of ['A', 5],
 Coxeter type of ['B', 1], Coxeter type of ['B', 5],
 Coxeter type of ['C', 1], Coxeter type of ['C', 5],
 Coxeter type of ['D', 4], Coxeter type of ['D', 5],
 Coxeter type of ['E', 6], Coxeter type of ['E', 7],
 Coxeter type of ['E', 8], Coxeter type of ['F', 4],
 Coxeter type of ['H', 3], Coxeter type of ['H', 4],
 Coxeter type of ['I', 10], Coxeter type of ['A', 2, 1],
 Coxeter type of ['B', 5, 1], Coxeter type of ['C', 5, 1],
 Coxeter type of ['D', 5, 1], Coxeter type of ['E', 6, 1],
 Coxeter type of ['E', 7, 1], Coxeter type of ['E', 8, 1],
 Coxeter type of ['F', 4, 1], Coxeter type of ['G', 2, 1],
 Coxeter type of ['A', 1, 1]]

>>> from sage.all import *
>>> CoxeterType.samples()
[Coxeter type of ['A', 1], Coxeter type of ['A', 5],
 Coxeter type of ['B', 1], Coxeter type of ['B', 5],
 Coxeter type of ['C', 1], Coxeter type of ['C', 5],
 Coxeter type of ['D', 4], Coxeter type of ['D', 5],
 Coxeter type of ['E', 6], Coxeter type of ['E', 7],
 Coxeter type of ['E', 8], Coxeter type of ['F', 4],
 Coxeter type of ['H', 3], Coxeter type of ['H', 4],
 Coxeter type of ['I', 10], Coxeter type of ['A', 2, 1],
 Coxeter type of ['B', 5, 1], Coxeter type of ['C', 5, 1],
 Coxeter type of ['D', 5, 1], Coxeter type of ['E', 6, 1],
 Coxeter type of ['E', 7, 1], Coxeter type of ['E', 8, 1],
 Coxeter type of ['F', 4, 1], Coxeter type of ['G', 2, 1],
 Coxeter type of ['A', 1, 1]]

The finite, affine and crystallographic options allow respectively for restricting to (non) finite, (non) affine, and (non) crystallographic Cartan types:

sage: CoxeterType.samples(finite=True)
[Coxeter type of ['A', 1], Coxeter type of ['A', 5],
 Coxeter type of ['B', 1], Coxeter type of ['B', 5],
 Coxeter type of ['C', 1], Coxeter type of ['C', 5],
 Coxeter type of ['D', 4], Coxeter type of ['D', 5],
 Coxeter type of ['E', 6], Coxeter type of ['E', 7],
 Coxeter type of ['E', 8], Coxeter type of ['F', 4],
 Coxeter type of ['H', 3], Coxeter type of ['H', 4],
 Coxeter type of ['I', 10]]

sage: CoxeterType.samples(affine=True)
[Coxeter type of ['A', 2, 1], Coxeter type of ['B', 5, 1],
 Coxeter type of ['C', 5, 1], Coxeter type of ['D', 5, 1],
 Coxeter type of ['E', 6, 1], Coxeter type of ['E', 7, 1],
 Coxeter type of ['E', 8, 1], Coxeter type of ['F', 4, 1],
 Coxeter type of ['G', 2, 1], Coxeter type of ['A', 1, 1]]

sage: CoxeterType.samples(crystallographic=True)
[Coxeter type of ['A', 1], Coxeter type of ['A', 5],
 Coxeter type of ['B', 1], Coxeter type of ['B', 5],
 Coxeter type of ['C', 1], Coxeter type of ['C', 5],
 Coxeter type of ['D', 4], Coxeter type of ['D', 5],
 Coxeter type of ['E', 6], Coxeter type of ['E', 7],
 Coxeter type of ['E', 8], Coxeter type of ['F', 4],
 Coxeter type of ['A', 2, 1], Coxeter type of ['B', 5, 1],
 Coxeter type of ['C', 5, 1], Coxeter type of ['D', 5, 1],
 Coxeter type of ['E', 6, 1], Coxeter type of ['E', 7, 1],
 Coxeter type of ['E', 8, 1], Coxeter type of ['F', 4, 1],
 Coxeter type of ['G', 2, 1], Coxeter type of ['A', 1, 1]]

sage: CoxeterType.samples(crystallographic=False)
[Coxeter type of ['H', 3],
 Coxeter type of ['H', 4],
 Coxeter type of ['I', 10]]

>>> from sage.all import *
>>> CoxeterType.samples(finite=True)
[Coxeter type of ['A', 1], Coxeter type of ['A', 5],
 Coxeter type of ['B', 1], Coxeter type of ['B', 5],
 Coxeter type of ['C', 1], Coxeter type of ['C', 5],
 Coxeter type of ['D', 4], Coxeter type of ['D', 5],
 Coxeter type of ['E', 6], Coxeter type of ['E', 7],
 Coxeter type of ['E', 8], Coxeter type of ['F', 4],
 Coxeter type of ['H', 3], Coxeter type of ['H', 4],
 Coxeter type of ['I', 10]]

>>> CoxeterType.samples(affine=True)
[Coxeter type of ['A', 2, 1], Coxeter type of ['B', 5, 1],
 Coxeter type of ['C', 5, 1], Coxeter type of ['D', 5, 1],
 Coxeter type of ['E', 6, 1], Coxeter type of ['E', 7, 1],
 Coxeter type of ['E', 8, 1], Coxeter type of ['F', 4, 1],
 Coxeter type of ['G', 2, 1], Coxeter type of ['A', 1, 1]]

>>> CoxeterType.samples(crystallographic=True)
[Coxeter type of ['A', 1], Coxeter type of ['A', 5],
 Coxeter type of ['B', 1], Coxeter type of ['B', 5],
 Coxeter type of ['C', 1], Coxeter type of ['C', 5],
 Coxeter type of ['D', 4], Coxeter type of ['D', 5],
 Coxeter type of ['E', 6], Coxeter type of ['E', 7],
 Coxeter type of ['E', 8], Coxeter type of ['F', 4],
 Coxeter type of ['A', 2, 1], Coxeter type of ['B', 5, 1],
 Coxeter type of ['C', 5, 1], Coxeter type of ['D', 5, 1],
 Coxeter type of ['E', 6, 1], Coxeter type of ['E', 7, 1],
 Coxeter type of ['E', 8, 1], Coxeter type of ['F', 4, 1],
 Coxeter type of ['G', 2, 1], Coxeter type of ['A', 1, 1]]

>>> CoxeterType.samples(crystallographic=False)
[Coxeter type of ['H', 3],
 Coxeter type of ['H', 4],
 Coxeter type of ['I', 10]]

Todo

add some reducible Coxeter types (suggestions?)

class sage.combinat.root_system.coxeter_type.CoxeterTypeFromCartanType(cartan_type)[source]

Bases: UniqueRepresentation, CoxeterType

A Coxeter type associated to a Cartan type.

cartan_type()[source]

Return the Cartan type used to construct self.

EXAMPLES:

sage: C = CoxeterType(['C',3])
sage: C.cartan_type()
['C', 3]

>>> from sage.all import *
>>> C = CoxeterType(['C',Integer(3)])
>>> C.cartan_type()
['C', 3]
component_types()[source]

A list of Coxeter types making up the reducible type.

EXAMPLES:

sage: CoxeterType(['A',2],['B',2]).component_types()
[Coxeter type of ['A', 2], Coxeter type of ['B', 2]]

sage: CoxeterType('A4xB3').component_types()
[Coxeter type of ['A', 4], Coxeter type of ['B', 3]]

sage: CoxeterType(['A', 2]).component_types()
Traceback (most recent call last):
...
ValueError: component types only defined for reducible types

>>> from sage.all import *
>>> CoxeterType(['A',Integer(2)],['B',Integer(2)]).component_types()
[Coxeter type of ['A', 2], Coxeter type of ['B', 2]]

>>> CoxeterType('A4xB3').component_types()
[Coxeter type of ['A', 4], Coxeter type of ['B', 3]]

>>> CoxeterType(['A', Integer(2)]).component_types()
Traceback (most recent call last):
...
ValueError: component types only defined for reducible types
coxeter_graph()[source]

Return the Coxeter graph of self.

EXAMPLES:

sage: C = CoxeterType(['H',3])
sage: C.coxeter_graph().edges(sort=True)                                    # needs sage.graphs
[(1, 2, 3), (2, 3, 5)]

>>> from sage.all import *
>>> C = CoxeterType(['H',Integer(3)])
>>> C.coxeter_graph().edges(sort=True)                                    # needs sage.graphs
[(1, 2, 3), (2, 3, 5)]
coxeter_matrix()[source]

Return the Coxeter matrix associated to self.

EXAMPLES:

sage: C = CoxeterType(['H',3])
sage: C.coxeter_matrix()                                                    # needs sage.graphs
[1 3 2]
[3 1 5]
[2 5 1]

>>> from sage.all import *
>>> C = CoxeterType(['H',Integer(3)])
>>> C.coxeter_matrix()                                                    # needs sage.graphs
[1 3 2]
[3 1 5]
[2 5 1]
index_set()[source]

Return the index set of self.

EXAMPLES:

sage: C = CoxeterType(['A', 4])
sage: C.index_set()
(1, 2, 3, 4)

>>> from sage.all import *
>>> C = CoxeterType(['A', Integer(4)])
>>> C.index_set()
(1, 2, 3, 4)
is_affine()[source]

Return if self is an affine type.

EXAMPLES:

sage: C = CoxeterType(['F', 4, 1])
sage: C.is_affine()
True

>>> from sage.all import *
>>> C = CoxeterType(['F', Integer(4), Integer(1)])
>>> C.is_affine()
True
is_crystallographic()[source]

Return if self is crystallographic.

EXAMPLES:

sage: C = CoxeterType(['C', 3])
sage: C.is_crystallographic()
True

sage: C = CoxeterType(['H', 3])
sage: C.is_crystallographic()
False

>>> from sage.all import *
>>> C = CoxeterType(['C', Integer(3)])
>>> C.is_crystallographic()
True

>>> C = CoxeterType(['H', Integer(3)])
>>> C.is_crystallographic()
False
is_finite()[source]

Return if self is a finite type.

EXAMPLES:

sage: C = CoxeterType(['E', 6])
sage: C.is_finite()
True

>>> from sage.all import *
>>> C = CoxeterType(['E', Integer(6)])
>>> C.is_finite()
True
is_irreducible()[source]

Return if self is irreducible.

EXAMPLES:

sage: C = CoxeterType(['A', 5])
sage: C.is_irreducible()
True

sage: C = CoxeterType('B3xB3')
sage: C.is_irreducible()
False

>>> from sage.all import *
>>> C = CoxeterType(['A', Integer(5)])
>>> C.is_irreducible()
True

>>> C = CoxeterType('B3xB3')
>>> C.is_irreducible()
False
is_reducible()[source]

Return if self is reducible.

EXAMPLES:

sage: C = CoxeterType(['A', 5])
sage: C.is_reducible()
False

sage: C = CoxeterType('A2xA2')
sage: C.is_reducible()
True

>>> from sage.all import *
>>> C = CoxeterType(['A', Integer(5)])
>>> C.is_reducible()
False

>>> C = CoxeterType('A2xA2')
>>> C.is_reducible()
True
is_simply_laced()[source]

Return if self is simply-laced.

EXAMPLES:

sage: C = CoxeterType(['A', 5])
sage: C.is_simply_laced()
True

sage: C = CoxeterType(['B', 3])
sage: C.is_simply_laced()
False

>>> from sage.all import *
>>> C = CoxeterType(['A', Integer(5)])
>>> C.is_simply_laced()
True

>>> C = CoxeterType(['B', Integer(3)])
>>> C.is_simply_laced()
False
rank()[source]

Return the rank of self.

EXAMPLES:

sage: C = CoxeterType(['I', 16])
sage: C.rank()
2

>>> from sage.all import *
>>> C = CoxeterType(['I', Integer(16)])
>>> C.rank()
2
relabel(relabelling)[source]

Return a relabelled copy of self.

EXAMPLES:

sage: ct = CoxeterType(['A',2])
sage: ct.relabel({1:-1, 2:-2})
Coxeter type of ['A', 2] relabelled by {1: -1, 2: -2}

>>> from sage.all import *
>>> ct = CoxeterType(['A',Integer(2)])
>>> ct.relabel({Integer(1):-Integer(1), Integer(2):-Integer(2)})
Coxeter type of ['A', 2] relabelled by {1: -1, 2: -2}
type()[source]

Return the type of self.

EXAMPLES:

sage: C = CoxeterType(['A', 4])
sage: C.type()
'A'

>>> from sage.all import *
>>> C = CoxeterType(['A', Integer(4)])
>>> C.type()
'A'