def __init__(self, coxeter_matrix, base_ring, index_set):
"""
Initialize ``self``.
EXAMPLES::
sage: W = CoxeterGroup([[1,3,2],[3,1,3],[2,3,1]])
sage: TestSuite(W).run() # long time
sage: W = CoxeterGroup([[1,3,2],[3,1,4],[2,4,1]], base_ring=QQbar)
sage: TestSuite(W).run() # long time
sage: W = CoxeterGroup([[1,3,2],[3,1,6],[2,6,1]])
sage: TestSuite(W).run(max_runs=30) # long time
sage: W = CoxeterGroup([[1,3,2],[3,1,-1],[2,-1,1]])
sage: TestSuite(W).run(max_runs=30) # long time
"""
self._matrix = coxeter_matrix
self._index_set = index_set
n = ZZ(coxeter_matrix.nrows())
MS = MatrixSpace(base_ring, n, sparse=True)
# FIXME: Hack because there is no ZZ \cup \{ \infty \}: -1 represents \infty
if base_ring is UniversalCyclotomicField():
val = lambda x: base_ring.gen(2*x) + ~base_ring.gen(2*x) if x != -1 else base_ring(2)
else:
from sage.functions.trig import cos
from sage.symbolic.constants import pi
val = lambda x: base_ring(2*cos(pi / x)) if x != -1 else base_ring(2)
gens = [MS.one() + MS({(i, j): val(coxeter_matrix[i, j])
for j in range(n)})
for i in range(n)]
FinitelyGeneratedMatrixGroup_generic.__init__(self, n, base_ring,
gens,
category=CoxeterGroups())
def super_categories(self):
r"""
EXAMPLES::
sage: WeylGroups().super_categories()
[Category of coxeter groups]
"""
return [CoxeterGroups()]
def super_categories(self):
r"""
EXAMPLES::
sage: FiniteCoxeterGroups().super_categories()
[Category of coxeter groups, Category of finite groups]
"""
return [CoxeterGroups(), FiniteGroups()]
def __classcall__(cls, W):
"""
EXAMPLES::
sage: from sage.monoids.j_trivial_monoids import *
sage: HeckeMonoid(['A',3]).cardinality()
24
"""
from sage.categories.coxeter_groups import CoxeterGroups
if not W in CoxeterGroups():
from sage.combinat.root_system.weyl_group import WeylGroup
W = WeylGroup(W) # CoxeterGroup(W)
return super(PiMonoid, cls).__classcall__(cls, W)
def _coerce_map_from_(self, P):
"""
Return ``True`` if ``P`` is a Coxeter group of the same
Coxeter type and ``False`` otherwise.
EXAMPLES::
sage: W = CoxeterGroup(["A",4])
sage: W2 = WeylGroup(["A",4])
sage: W._coerce_map_from_(W2)
True
sage: W3 = WeylGroup(["A",4], implementation="permutation")
sage: W._coerce_map_from_(W3)
True
sage: W4 = WeylGroup(["A",3])
sage: W.has_coerce_map_from(W4)
False
"""
if P in CoxeterGroups() and P.coxeter_type() is self.coxeter_type():
return True
return super(CoxeterMatrixGroup, self)._coerce_map_from_(P)
def __classcall_private__(cls, data):
r"""
EXAMPLES::
sage: from sage.combinat.fully_commutative_elements import FullyCommutativeElements
sage: x1 = FullyCommutativeElements(CoxeterGroup(['B', 3])); x1
Fully commutative elements of Finite Coxeter group over Number Field in a with defining polynomial x^2 - 2 with a = 1.414213562373095? with Coxeter matrix:
[1 3 2]
[3 1 4]
[2 4 1]
sage: x2 = FullyCommutativeElements(['B', 3]); x2
Fully commutative elements of Finite Coxeter group over Number Field in a with defining polynomial x^2 - 2 with a = 1.414213562373095? with Coxeter matrix:
[1 3 2]
[3 1 4]
[2 4 1]
sage: x3 = FullyCommutativeElements(CoxeterMatrix([[1, 3, 2], [3, 1, 4], [2, 4, 1]])); x3
Fully commutative elements of Finite Coxeter group over Number Field in a with defining polynomial x^2 - 2 with a = 1.414213562373095? with Coxeter matrix:
[1 3 2]
[3 1 4]
[2 4 1]
sage: x1 is x2 is x3
True
sage: FullyCommutativeElements(CartanType(['B', 3]).relabel({1: 3, 2: 2, 3: 1}))
Fully commutative elements of Finite Coxeter group over Number Field in a with defining polynomial x^2 - 2 with a = 1.414213562373095? with Coxeter matrix:
[1 4 2]
[4 1 3]
[2 3 1]
sage: m = CoxeterMatrix([(1, 5, 2, 2, 2), (5, 1, 3, 2, 2), (2, 3, 1, 3, 2), (2, 2, 3, 1, 3), (2, 2, 2, 3, 1)]); FullyCommutativeElements(m)
Fully commutative elements of Coxeter group over Universal Cyclotomic Field with Coxeter matrix:
[1 5 2 2 2]
[5 1 3 2 2]
[2 3 1 3 2]
[2 2 3 1 3]
[2 2 2 3 1]
"""
if data in CoxeterGroups():
group = data
else:
group = CoxeterGroup(data)
return super(cls, FullyCommutativeElements).__classcall__(cls, group)
def __init__(self, coxeter_matrix, base_ring, index_set):
"""
Initialize ``self``.
EXAMPLES::
sage: W = CoxeterGroup([[1,3,2],[3,1,3],[2,3,1]])
sage: TestSuite(W).run() # long time
sage: W = CoxeterGroup([[1,3,2],[3,1,4],[2,4,1]], base_ring=QQbar)
sage: TestSuite(W).run() # long time
sage: W = CoxeterGroup([[1,3,2],[3,1,6],[2,6,1]])
sage: TestSuite(W).run(max_runs=30) # long time
sage: W = CoxeterGroup([[1,3,2],[3,1,-1],[2,-1,1]])
sage: TestSuite(W).run(max_runs=30) # long time
We check that :trac:`16630` is fixed::
sage: CoxeterGroup(['D',4], base_ring=QQ).category()
Category of finite coxeter groups
sage: CoxeterGroup(['H',4], base_ring=QQbar).category()
Category of finite coxeter groups
sage: F = CoxeterGroups().Finite()
sage: all(CoxeterGroup([letter,i]) in F
....: for i in range(2,5) for letter in ['A','B','D'])
True
sage: all(CoxeterGroup(['E',i]) in F for i in range(6,9))
True
sage: CoxeterGroup(['F',4]).category()
Category of finite coxeter groups
sage: CoxeterGroup(['G',2]).category()
Category of finite coxeter groups
sage: all(CoxeterGroup(['H',i]) in F for i in range(3,5))
True
sage: all(CoxeterGroup(['I',i]) in F for i in range(2,5))
True
"""
self._matrix = coxeter_matrix
n = coxeter_matrix.rank()
# Compute the matrix with entries `2 \cos( \pi / m_{ij} )`.
MS = MatrixSpace(base_ring, n, sparse=True)
MC = MS._get_matrix_class()
# FIXME: Hack because there is no ZZ \cup \{ \infty \}: -1 represents \infty
E = UniversalCyclotomicField().gen
if base_ring is UniversalCyclotomicField():
def val(x):
if x == -1:
return 2
else:
return E(2 * x) + ~E(2 * x)
elif is_QuadraticField(base_ring):
def val(x):
if x == -1:
return 2
else:
return base_ring(
(E(2 * x) + ~E(2 * x)).to_cyclotomic_field())
else:
from sage.functions.trig import cos
from sage.symbolic.constants import pi
def val(x):
if x == -1:
return 2
else:
return base_ring(2 * cos(pi / x))
gens = [
MS.one() +
MC(MS,
entries={(i, j): val(coxeter_matrix[index_set[i], index_set[j]])
for j in range(n)},
coerce=True,
copy=True) for i in range(n)
]
# Make the generators dense matrices for consistency and speed
gens = [g.dense_matrix() for g in gens]
category = CoxeterGroups()
# Now we shall see if the group is finite, and, if so, refine
# the category to ``category.Finite()``. Otherwise the group is
# infinite and we refine the category to ``category.Infinite()``.
if self._matrix.is_finite():
category = category.Finite()
else:
category = category.Infinite()
self._index_set_inverse = {
i: ii
for ii, i in enumerate(self._matrix.index_set())
}
FinitelyGeneratedMatrixGroup_generic.__init__(self,
ZZ(n),
base_ring,
gens,
category=category)
def __init__(self, coxeter_matrix, base_ring, index_set):
"""
Initialize ``self``.
EXAMPLES::
sage: W = CoxeterGroup([[1,3,2],[3,1,3],[2,3,1]])
sage: TestSuite(W).run() # long time
sage: W = CoxeterGroup([[1,3,2],[3,1,4],[2,4,1]], base_ring=QQbar)
sage: TestSuite(W).run() # long time
sage: W = CoxeterGroup([[1,3,2],[3,1,6],[2,6,1]])
sage: TestSuite(W).run(max_runs=30) # long time
sage: W = CoxeterGroup([[1,3,2],[3,1,-1],[2,-1,1]])
sage: TestSuite(W).run(max_runs=30) # long time
We check that :trac:`16630` is fixed::
sage: CoxeterGroup(['D',4], base_ring=QQ).category()
Category of finite coxeter groups
sage: CoxeterGroup(['H',4], base_ring=QQbar).category()
Category of finite coxeter groups
sage: F = CoxeterGroups().Finite()
sage: all(CoxeterGroup([letter,i]) in F
....: for i in range(2,5) for letter in ['A','B','D'])
True
sage: all(CoxeterGroup(['E',i]) in F for i in range(6,9))
True
sage: CoxeterGroup(['F',4]).category()
Category of finite coxeter groups
sage: CoxeterGroup(['G',2]).category()
Category of finite coxeter groups
sage: all(CoxeterGroup(['H',i]) in F for i in range(3,5))
True
sage: all(CoxeterGroup(['I',i]) in F for i in range(2,5))
True
"""
self._matrix = coxeter_matrix
self._index_set = index_set
n = ZZ(coxeter_matrix.nrows())
# Compute the matrix with entries `2 \cos( \pi / m_{ij} )`.
MS = MatrixSpace(base_ring, n, sparse=True)
MC = MS._get_matrix_class()
# FIXME: Hack because there is no ZZ \cup \{ \infty \}: -1 represents \infty
if base_ring is UniversalCyclotomicField():
val = lambda x: base_ring.gen(2 * x) + ~base_ring.gen(
2 * x) if x != -1 else base_ring(2)
else:
from sage.functions.trig import cos
from sage.symbolic.constants import pi
val = lambda x: base_ring(2 * cos(pi / x)
) if x != -1 else base_ring(2)
gens = [
MS.one() + MC(MS,
entries={(i, j): val(coxeter_matrix[i, j])
for j in range(n)},
coerce=True,
copy=True) for i in range(n)
]
# Compute the matrix with entries `- \cos( \pi / m_{ij} )`.
# This describes the bilinear form corresponding to this
# Coxeter system, and might lead us out of our base ring.
base_field = base_ring.fraction_field()
MS2 = MatrixSpace(base_field, n, sparse=True)
MC2 = MS2._get_matrix_class()
self._bilinear = MC2(MS2,
entries={
(i, j):
val(coxeter_matrix[i, j]) / base_field(-2)
for i in range(n) for j in range(n)
if coxeter_matrix[i, j] != 2
},
coerce=True,
copy=True)
self._bilinear.set_immutable()
category = CoxeterGroups()
# Now we shall see if the group is finite, and, if so, refine
# the category to ``category.Finite()``. Otherwise the group is
# infinite and we refine the category to ``category.Infinite()``.
is_finite = self._finite_recognition()
if is_finite:
category = category.Finite()
else:
category = category.Infinite()
FinitelyGeneratedMatrixGroup_generic.__init__(self,
n,
base_ring,
gens,
category=category)
