def __classcall_private__(cls,
data=None,
index_set=None,
cartan_type=None,
cartan_type_check=True,
borcherds=None):
"""
Normalize input so we can inherit from sparse integer matrix.
.. NOTE::
To disable the Cartan type check, use the optional argument
``cartan_type_check = False``.
EXAMPLES::
sage: C = CartanMatrix(['A',1,1])
sage: C2 = CartanMatrix([[2, -2], [-2, 2]])
sage: C3 = CartanMatrix(matrix([[2, -2], [-2, 2]]), [0, 1])
sage: C == C2 and C == C3
True
TESTS:
Check that :trac:`15740` is fixed::
sage: d = DynkinDiagram()
sage: d.add_edge('a', 'b', 2)
sage: d.index_set()
('a', 'b')
sage: cm = CartanMatrix(d)
sage: cm.index_set()
('a', 'b')
"""
# Special case with 0 args and kwds has Cartan type
if cartan_type is not None and data is None:
data = CartanType(cartan_type)
if data is None:
data = []
n = 0
index_set = tuple()
cartan_type = None
subdivisions = None
elif isinstance(data, CartanMatrix):
if index_set is not None:
d = {a: index_set[i] for i, a in enumerate(data.index_set())}
return data.relabel(d)
return data
else:
dynkin_diagram = None
subdivisions = None
from sage.combinat.root_system.dynkin_diagram import DynkinDiagram_class
if isinstance(data, DynkinDiagram_class):
dynkin_diagram = data
cartan_type = data._cartan_type
else:
try:
cartan_type = CartanType(data)
dynkin_diagram = cartan_type.dynkin_diagram()
except (TypeError, ValueError):
pass
if dynkin_diagram is not None:
n = dynkin_diagram.rank()
index_set = dynkin_diagram.index_set()
oir = dynkin_diagram.odd_isotropic_roots()
reverse = {a: i for i, a in enumerate(index_set)}
if isinstance(borcherds, (list, tuple)):
if (len(borcherds) != len(index_set) and
not all(val in ZZ and (val == 2 or
(val % 2 == 0 and val < 0))
for val in borcherds)):
raise ValueError(
"the input data is not a Borcherds-Cartan matrix")
data = {(i, i): val if index_set[i] not in oir else 0
for i, val in enumerate(borcherds)}
else:
data = {(i, i): 2 if index_set[i] not in oir else 0
for i in range(n)}
for (i, j, l) in dynkin_diagram.edge_iterator():
data[(reverse[j], reverse[i])] = -l
else:
M = matrix(data)
if borcherds:
if not is_borcherds_cartan_matrix(M):
raise ValueError(
"the input matrix is not a Borcherds-Cartan matrix"
)
else:
if not is_generalized_cartan_matrix(M):
raise ValueError(
"the input matrix is not a generalized Cartan matrix"
)
n = M.ncols()
data = M.dict()
subdivisions = M._subdivisions
if index_set is None:
index_set = tuple(range(n))
else:
index_set = tuple(index_set)
if len(index_set) != n and len(set(index_set)) != n:
raise ValueError("the given index set is not valid")
# We can do the Cartan type initialization later as this is not
# a unique representation
mat = typecall(cls, MatrixSpace(ZZ, n, sparse=True), data, False, True)
# FIXME: We have to initialize the CartanMatrix part separately because
# of the __cinit__ of the matrix. We should get rid of this workaround
mat._CM_init(cartan_type, index_set, cartan_type_check)
mat._subdivisions = subdivisions
return mat
def __classcall_private__(cls,
data=None,
index_set=None,
coxeter_type=None,
cartan_type=None,
coxeter_type_check=True):
r"""
A Coxeter matrix can we created via a graph, a Coxeter type, or
a matrix.
.. NOTE::
To disable the Coxeter type check, use the optional argument
``coxeter_type_check = False``.
EXAMPLES::
sage: C = CoxeterMatrix(['A',1,1],['a','b'])
sage: C2 = CoxeterMatrix([[1, -1], [-1, 1]])
sage: C3 = CoxeterMatrix(matrix([[1, -1], [-1, 1]]), [0, 1])
sage: C == C2 and C == C3
True
Check with `\infty` because of the hack of using `-1` to represent
`\infty` in the Coxeter matrix::
sage: G = Graph([(0, 1, 3), (1, 2, oo)])
sage: W1 = CoxeterMatrix([[1, 3, 2], [3, 1, -1], [2, -1, 1]])
sage: W2 = CoxeterMatrix(G)
sage: W1 == W2
True
sage: CoxeterMatrix(W1.coxeter_graph()) == W1
True
The base ring of the matrix depends on the entries given::
sage: CoxeterMatrix([[1,-1],[-1,1]])._matrix.base_ring()
Integer Ring
sage: CoxeterMatrix([[1,-3/2],[-3/2,1]])._matrix.base_ring()
Rational Field
sage: CoxeterMatrix([[1,-1.5],[-1.5,1]])._matrix.base_ring()
Real Field with 53 bits of precision
"""
if not data:
if coxeter_type:
data = CoxeterType(coxeter_type)
elif cartan_type:
data = CoxeterType(CartanType(cartan_type))
# Special cases with no arguments passed
if not data:
data = []
n = 0
index_set = tuple()
coxeter_type = None
base_ring = ZZ
mat = typecall(cls, MatrixSpace(base_ring, n, sparse=False), data,
coxeter_type, index_set)
mat._subdivisions = None
return mat
if isinstance(data, CoxeterMatrix): # Initiate from itself
return data
# Initiate from a graph:
# TODO: Check if a CoxeterDiagram once implemented
if isinstance(data, Graph):
return cls._from_graph(data, coxeter_type_check)
# Get the Coxeter type
coxeter_type = None
from sage.combinat.root_system.cartan_type import CartanType_abstract
if isinstance(data, CartanType_abstract):
coxeter_type = data.coxeter_type()
else:
try:
coxeter_type = CoxeterType(data)
except (TypeError, ValueError, NotImplementedError):
pass
# Initiate from a Coxeter type
if coxeter_type:
return cls._from_coxetertype(coxeter_type)
# TODO:: remove when oo is possible in matrices.
n = len(data[0])
data = [x if x != infinity else -1 for r in data for x in r]
data = matrix(n, n, data)
# until here
# Get the index set
if index_set:
index_set = tuple(index_set)
else:
index_set = tuple(range(1, n + 1))
if len(set(index_set)) != n:
raise ValueError("the given index set is not valid")
return cls._from_matrix(data, coxeter_type, index_set,
coxeter_type_check)
def find_cartan_type_from_matrix(CM):
r"""
Find a Cartan type by direct comparison of Dynkin diagrams given from
the generalized Cartan matrix ``CM`` and return ``None`` if not found.
INPUT:
- ``CM`` -- a generalized Cartan matrix
EXAMPLES::
sage: from sage.combinat.root_system.cartan_matrix import find_cartan_type_from_matrix
sage: CM = CartanMatrix([[2,-1,-1], [-1,2,-1], [-1,-1,2]])
sage: find_cartan_type_from_matrix(CM)
['A', 2, 1]
sage: CM = CartanMatrix([[2,-1,0], [-1,2,-2], [0,-1,2]])
sage: find_cartan_type_from_matrix(CM)
['C', 3] relabelled by {1: 0, 2: 1, 3: 2}
sage: CM = CartanMatrix([[2,-1,-2], [-1,2,-1], [-2,-1,2]])
sage: find_cartan_type_from_matrix(CM)
"""
types = []
for S in CM.dynkin_diagram().connected_components_subgraphs():
S = DiGraph(S) # We need a simple digraph here
n = S.num_verts()
# Build the list to test based upon rank
if n == 1:
types.append(CartanType(['A', 1]))
continue
test = [['A', n]]
if n >= 2:
if n == 2:
test += [['G', 2], ['A', 2, 2]]
test += [['B', n], ['A', n - 1, 1]]
if n >= 3:
if n == 3:
test.append(['G', 2, 1])
test += [['C', n], ['BC', n - 1, 2], ['C', n - 1, 1]]
if n >= 4:
if n == 4:
test.append(['F', 4])
test += [['D', n], ['B', n - 1, 1]]
if n >= 5:
if n == 5:
test.append(['F', 4, 1])
test.append(['D', n - 1, 1])
if n == 6:
test.append(['E', 6])
elif n == 7:
test += [['E', 7], ['E', 6, 1]]
elif n == 8:
test += [['E', 8], ['E', 7, 1]]
elif n == 9:
test.append(['E', 8, 1])
# Test every possible Cartan type and its dual
found = False
for x in test:
ct = CartanType(x)
T = DiGraph(ct.dynkin_diagram()) # We need a simple digraph here
iso, match = T.is_isomorphic(S, certificate=True, edge_labels=True)
if iso:
types.append(ct.relabel(match))
found = True
break
if ct == ct.dual():
continue # self-dual, so nothing more to test
ct = ct.dual()
T = DiGraph(ct.dynkin_diagram()) # We need a simple digraph here
iso, match = T.is_isomorphic(S, certificate=True, edge_labels=True)
if iso:
types.append(ct.relabel(match))
found = True
break
if not found:
return None
return CartanType(types)
def Associahedron(cartan_type, backend='ppl'):
r"""
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:`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
:class:`sage.combinat.root_system.cartan_type.CartanTypeFactory`
- ``backend`` -- string (``'ppl'``); the backend to use;
see :meth:`sage.geometry.polyhedron.constructor.Polyhedron`
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
TESTS::
sage: sorted(polytopes.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(polytopes.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: polytopes.associahedron(['A',4]).f_vector()
(1, 42, 84, 56, 14, 1)
sage: polytopes.associahedron(['B',4]).f_vector()
(1, 70, 140, 90, 20, 1)
sage: p1 = polytopes.associahedron(['A',4], backend='normaliz') # optional - pynormaliz
sage: TestSuite(p1).run(skip='_test_pickling') # optional - pynormaliz
sage: p2 = polytopes.associahedron(['A',4], backend='cdd')
sage: TestSuite(p2).run()
sage: p3 = polytopes.associahedron(['A',4], backend='field')
sage: TestSuite(p3).run()
"""
cartan_type = CartanType(cartan_type)
parent = Associahedra(QQ, cartan_type.rank(), backend)
return parent(cartan_type)
def DynkinDiagram(*args, **kwds):
r"""
Return the Dynkin diagram corresponding to the input.
INPUT:
The input can be one of the following:
- empty to obtain an empty Dynkin diagram
- a Cartan type
- a Cartan matrix
- a Cartan matrix and an indexing set
Also one can input an index_set by
The edge multiplicities are encoded as edge labels. This uses the
convention in Hong and Kang, Kac, Fulton Harris, and crystals. This is the
**opposite** convention in Bourbaki and Wikipedia's Dynkin diagram
(:wikipedia:`Dynkin_diagram`). That is for `i \neq j`::
i <--k-- j <==> a_ij = -k
<==> -scalar(coroot[i], root[j]) = k
<==> multiple arrows point from the longer root
to the shorter one
For example, in type `C_2`, we have::
sage: C2 = DynkinDiagram(['C',2]); C2
O=<=O
1 2
C2
sage: C2.cartan_matrix()
[ 2 -2]
[-1 2]
However Bourbaki would have the Cartan matrix as:
.. MATH::
\begin{bmatrix}
2 & -1 \\
-2 & 2
\end{bmatrix}.
EXAMPLES::
sage: DynkinDiagram(['A', 4])
O---O---O---O
1 2 3 4
A4
sage: DynkinDiagram(['A',1],['A',1])
O
1
O
2
A1xA1
sage: R = RootSystem("A2xB2xF4")
sage: DynkinDiagram(R)
O---O
1 2
O=>=O
3 4
O---O=>=O---O
5 6 7 8
A2xB2xF4
sage: R = RootSystem("A2xB2xF4")
sage: CM = R.cartan_matrix(); CM
[ 2 -1| 0 0| 0 0 0 0]
[-1 2| 0 0| 0 0 0 0]
[-----+-----+-----------]
[ 0 0| 2 -1| 0 0 0 0]
[ 0 0|-2 2| 0 0 0 0]
[-----+-----+-----------]
[ 0 0| 0 0| 2 -1 0 0]
[ 0 0| 0 0|-1 2 -1 0]
[ 0 0| 0 0| 0 -2 2 -1]
[ 0 0| 0 0| 0 0 -1 2]
sage: DD = DynkinDiagram(CM); DD
O---O
1 2
O=>=O
3 4
O---O=>=O---O
5 6 7 8
A2xB2xF4
sage: DD.cartan_matrix()
[ 2 -1 0 0 0 0 0 0]
[-1 2 0 0 0 0 0 0]
[ 0 0 2 -1 0 0 0 0]
[ 0 0 -2 2 0 0 0 0]
[ 0 0 0 0 2 -1 0 0]
[ 0 0 0 0 -1 2 -1 0]
[ 0 0 0 0 0 -2 2 -1]
[ 0 0 0 0 0 0 -1 2]
We can also create Dynkin diagrams from arbitrary Cartan matrices::
sage: C = CartanMatrix([[2, -3], [-4, 2]])
sage: DynkinDiagram(C)
Dynkin diagram of rank 2
sage: C.index_set()
(0, 1)
sage: CI = CartanMatrix([[2, -3], [-4, 2]], [3, 5])
sage: DI = DynkinDiagram(CI)
sage: DI.index_set()
(3, 5)
sage: CII = CartanMatrix([[2, -3], [-4, 2]])
sage: DII = DynkinDiagram(CII, ('y', 'x'))
sage: DII.index_set()
('x', 'y')
.. SEEALSO::
:func:`CartanType` for a general discussion on Cartan
types and in particular node labeling conventions.
TESTS:
Check that :trac:`15277` is fixed by not having edges from 0's::
sage: CM = CartanMatrix([[2,-1,0,0],[-3,2,-2,-2],[0,-1,2,-1],[0,-1,-1,2]])
sage: CM
[ 2 -1 0 0]
[-3 2 -2 -2]
[ 0 -1 2 -1]
[ 0 -1 -1 2]
sage: CM.dynkin_diagram().edges()
[(0, 1, 3),
(1, 0, 1),
(1, 2, 1),
(1, 3, 1),
(2, 1, 2),
(2, 3, 1),
(3, 1, 2),
(3, 2, 1)]
"""
if len(args) == 0:
return DynkinDiagram_class()
mat = args[0]
if is_Matrix(mat):
mat = CartanMatrix(*args)
if isinstance(mat, CartanMatrix):
if mat.cartan_type() is not mat:
try:
return mat.cartan_type().dynkin_diagram()
except AttributeError:
ct = CartanType(*args)
raise ValueError(
"Dynkin diagram data not yet hardcoded for type %s" % ct)
if len(args) > 1:
index_set = tuple(args[1])
elif "index_set" in kwds:
index_set = tuple(kwds["index_set"])
else:
index_set = mat.index_set()
D = DynkinDiagram_class(index_set=index_set)
for (i, j) in mat.nonzero_positions():
if i != j:
D.add_edge(index_set[i], index_set[j], -mat[j, i])
return D
ct = CartanType(*args)
try:
return ct.dynkin_diagram()
except AttributeError:
raise ValueError("Dynkin diagram data not yet hardcoded for type %s" %
ct)
def find_cartan_type_from_matrix(CM):
"""
Find a Cartan type by direct comparison of matrices given from the
generalized Cartan matrix ``CM`` and return ``None`` if not found.
INPUT:
- ``CM`` -- A generalized Cartan matrix
EXAMPLES::
sage: from sage.combinat.root_system.cartan_matrix import find_cartan_type_from_matrix
sage: M = matrix([[2,-1,-1], [-1,2,-1], [-1,-1,2]])
sage: find_cartan_type_from_matrix(M)
['A', 2, 1]
sage: M = matrix([[2,-1,0], [-1,2,-2], [0,-1,2]])
sage: find_cartan_type_from_matrix(M)
['C', 3]
sage: M = matrix([[2,-1,-2], [-1,2,-1], [-2,-1,2]])
sage: find_cartan_type_from_matrix(M)
"""
n = CM.ncols()
# Build the list to test based upon rank
if n == 1:
return CartanType(['A', 1])
test = [['A', n]]
if n >= 2:
if n == 2:
test += [['G', 2], ['A', 2, 2]]
test += [['B', n], ['A', n - 1, 1]]
if n >= 3:
if n == 3:
test += [['G', 2, 1], ['D', 4, 3]]
test += [['C', n], ['BC', n - 1, 2], ['C', n - 1, 1]]
if n >= 4:
if n == 4:
test += [['F', 4], ['G', 2, 1], ['D', 4, 3]]
test += [['D', n], ['B', n - 1, 1]]
if n == 5:
test += [['F', 4, 1], ['D', n - 1, 1]]
elif n == 6:
test.append(['E', 6])
elif n == 7:
test += [['E', 7], ['E', 6, 1]]
elif n == 8:
test += [['E', 8], ['E', 7, 1]]
elif n == 9:
test.append(['E', 8, 1])
# Test every possible Cartan type and its dual
for x in test:
ct = CartanType(x)
if ct.cartan_matrix() == CM:
return ct
if ct == ct.dual():
continue
ct = ct.dual()
if ct.cartan_matrix() == CM:
return ct
return None
def HighestWeightCrystal(dominant_weight):
r"""
Returns an implementation of the highest weight crystal of highest weight `dominant_weight`.
This is currently only implemented for crystals of type `E_6` and `E_7`.
TODO: implement highest weight crystals for classical types `A_n`, `B_n`, `C_n`, `D_n` using tableaux.
EXAMPLES::
sage: C=CartanType(['E',6])
sage: La=C.root_system().weight_lattice().fundamental_weights()
sage: T = HighestWeightCrystal(La[1])
sage: T.cardinality()
27
sage: T = HighestWeightCrystal(La[6])
sage: T.cardinality()
27
sage: T = HighestWeightCrystal(La[2])
sage: T.cardinality()
78
sage: T = HighestWeightCrystal(La[4])
sage: T.cardinality()
2925
sage: T = HighestWeightCrystal(La[3])
sage: T.cardinality()
351
sage: T = HighestWeightCrystal(La[5])
sage: T.cardinality()
351
sage: C=CartanType(['E',7])
sage: La=C.root_system().weight_lattice().fundamental_weights()
sage: T = HighestWeightCrystal(La[1])
sage: T.cardinality()
133
sage: T = HighestWeightCrystal(La[2])
sage: T.cardinality()
912
sage: T = HighestWeightCrystal(La[3])
sage: T.cardinality()
8645
sage: T = HighestWeightCrystal(La[4])
sage: T.cardinality()
365750
sage: T = HighestWeightCrystal(La[5])
sage: T.cardinality()
27664
sage: T = HighestWeightCrystal(La[6])
sage: T.cardinality()
1539
sage: T = HighestWeightCrystal(La[7])
sage: T.cardinality()
56
sage: C = CartanType(['C',2,1])
sage: La = C.root_system().weight_lattice().fundamental_weights()
sage: T = HighestWeightCrystal(La[1])
sage: [p for p in T.subcrystal(max_depth=3)]
[(Lambda[1],), (Lambda[0] - Lambda[1] + Lambda[2],), (-Lambda[0] + Lambda[1] + Lambda[2] - delta,),
(Lambda[0] + Lambda[1] - Lambda[2],), (-Lambda[0] + 3*Lambda[1] - Lambda[2] - delta,), (2*Lambda[0] - Lambda[1],),
(-Lambda[1] + 2*Lambda[2] - delta,)]
"""
cartan_type = dominant_weight.parent().cartan_type()
if cartan_type.is_finite() and cartan_type.type() in ['A','B','C','D']:
raise NotImplementedError
elif cartan_type == CartanType(['E',6]):
return FiniteDimensionalHighestWeightCrystal_TypeE6(dominant_weight)
elif cartan_type == CartanType(['E',7]):
return FiniteDimensionalHighestWeightCrystal_TypeE7(dominant_weight)
elif cartan_type.is_affine():
return CrystalOfLSPaths(cartan_type,[dominant_weight[i] for i in cartan_type.index_set()])
else:
raise NotImplementedError
def TensorProductOfCrystals(*crystals, **options):
r"""
Tensor product of crystals.
Given two crystals `B` and `B'` of the same type,
one can form the tensor product `B \otimes B'`. As a set
`B \otimes B'` is the Cartesian product
`B \times B'`. The crystal operators `f_i` and
`e_i` act on `b \otimes b' \in B \otimes B'` as
follows:
.. math::
f_i(b \otimes b') = \begin{cases}
f_i(b) \otimes b' & \text{if $\varepsilon_i(b) \ge \varphi_i(b')$}\\
b \otimes f_i(b') & \text{otherwise}
\end{cases}
and
.. math::
e_i(b \otimes b') = \begin{cases}
b \otimes e_i(b') & \text{if $\varepsilon_i(b) \le \varphi_i(b')$}\\
e_i(b) \otimes b' & \text{otherwise.}
\end{cases}
Note that this is the opposite of Kashiwara's convention for tensor
products of crystals.
EXAMPLES:
We construct the type `A_2`-crystal generated by `2 \otimes 1 \otimes 1`::
sage: C = CrystalOfLetters(['A',2])
sage: T = TensorProductOfCrystals(C,C,C,generators=[[C(2),C(1),C(1)]])
It has `8` elements::
sage: T.list()
[[2, 1, 1], [2, 1, 2], [2, 1, 3], [3, 1, 3], [3, 2, 3], [3, 1, 1], [3, 1, 2], [3, 2, 2]]
One can also check the Cartan type of the crystal::
sage: T.cartan_type()
['A', 2]
Other examples include crystals of tableaux (which internally are represented as tensor products
obtained by reading the tableaux columnwise)::
sage: C = CrystalOfTableaux(['A',3], shape=[1,1,0])
sage: D = CrystalOfTableaux(['A',3], shape=[1,0,0])
sage: T = TensorProductOfCrystals(C,D, generators=[[C(rows=[[1], [2]]), D(rows=[[1]])], [C(rows=[[2], [3]]), D(rows=[[1]])]])
sage: T.cardinality()
24
sage: TestSuite(T).run()
sage: T.module_generators
[[[[1], [2]], [[1]]], [[[2], [3]], [[1]]]]
sage: [x.weight() for x in T.module_generators]
[(2, 1, 0, 0), (1, 1, 1, 0)]
If no module generators are specified, we obtain the full tensor
product::
sage: C=CrystalOfLetters(['A',2])
sage: T=TensorProductOfCrystals(C,C)
sage: T.list()
[[1, 1], [1, 2], [1, 3], [2, 1], [2, 2], [2, 3], [3, 1], [3, 2], [3, 3]]
sage: T.cardinality()
9
For a tensor product of crystals without module generators, the
default implementation of module_generators contains all elements
in the tensor product of the crystals. If there is a subset of
elements in the tensor product that still generates the crystal,
this needs to be implemented for the specific crystal separately::
sage: T.module_generators.list()
[[1, 1], [1, 2], [1, 3], [2, 1], [2, 2], [2, 3], [3, 1], [3, 2], [3, 3]]
For classical highest weight crystals, it is also possible to list
all highest weight elements::
sage: C = CrystalOfLetters(['A',2])
sage: T = TensorProductOfCrystals(C,C,C,generators=[[C(2),C(1),C(1)],[C(1),C(2),C(1)]])
sage: T.highest_weight_vectors()
[[2, 1, 1], [1, 2, 1]]
"""
crystals = tuple(crystals)
if "cartan_type" in options:
cartan_type = CartanType(options["cartan_type"])
else:
if len(crystals) == 0:
raise ValueError, "you need to specify the Cartan type if the tensor product list is empty"
else:
cartan_type = crystals[0].cartan_type()
if "generators" in options:
generators = tuple(tuple(x) if isinstance(x, list) else x for x in options["generators"])
return TensorProductOfCrystalsWithGenerators(crystals, generators=generators, cartan_type = cartan_type)
else:
return FullTensorProductOfCrystals(crystals, cartan_type = cartan_type)
def CoxeterGroup(cartan_type, implementation=None):
"""
INPUT:
- ``cartan_type`` -- a cartan type (or coercible into; see :class:`CartanType`)
- ``implementation`` -- "permutation", "matrix", "coxeter3", or None (default: None)
Returns an implementation of the Coxeter group of type
``cartan_type``.
EXAMPLES:
If ``implementation`` is not specified, a permutation
representation is returned whenever possible (finite irreducible
Cartan type, with the GAP3 Chevie package available)::
sage: W = CoxeterGroup(["A",2])
sage: W # optional - chevie
Permutation Group with generators [(1,3)(2,5)(4,6), (1,4)(2,3)(5,6)]
Otherwise, a Weyl group is returned::
sage: W = CoxeterGroup(["A",3,1])
sage: W
Weyl Group of type ['A', 3, 1] (as a matrix group acting on the root space)
We now use the ``implementation`` option::
sage: W = CoxeterGroup(["A",2], implementation = "permutation") # optional - chevie
sage: W # optional - chevie
Permutation Group with generators [(1,3)(2,5)(4,6), (1,4)(2,3)(5,6)]
sage: W.category() # optional - chevie
Join of Category of finite permutation groups and Category of finite coxeter groups
sage: W = CoxeterGroup(["A",2], implementation = "matrix")
sage: W
Weyl Group of type ['A', 2] (as a matrix group acting on the ambient space)
sage: W = CoxeterGroup(["H",3], implementation = "matrix")
Traceback (most recent call last):
...
NotImplementedError: Coxeter group of type ['H', 3] as matrix group not implemented
sage: W = CoxeterGroup(["A",4,1], implementation = "permutation")
Traceback (most recent call last):
...
NotImplementedError: Coxeter group of type ['A', 4, 1] as permutation group not implemented
"""
assert implementation in ["permutation", "matrix", "coxeter3", None]
cartan_type = CartanType(cartan_type)
if implementation is None:
if cartan_type.is_finite() and cartan_type.is_irreducible(
) and is_chevie_available():
implementation = "permutation"
else:
implementation = "matrix"
if implementation == "coxeter3":
try:
from sage.libs.coxeter3.coxeter_group import CoxeterGroup
except ImportError:
raise RuntimeError, "coxeter3 must be installed"
else:
return CoxeterGroup(cartan_type)
if implementation == "permutation" and is_chevie_available() and \
cartan_type.is_finite() and cartan_type.is_irreducible():
return CoxeterGroupAsPermutationGroup(cartan_type)
elif implementation == "matrix" and cartan_type.is_crystalographic():
return WeylGroup(cartan_type)
else:
raise NotImplementedError, "Coxeter group of type %s as %s group not implemented " % (
cartan_type, implementation)
def __classcall_private__(cls, *crystals, **options):
"""
Create the correct parent object.
EXAMPLES::
sage: C = crystals.Letters(['A',2])
sage: T = crystals.TensorProduct(C, C)
sage: T2 = crystals.TensorProduct(C, C, cartan_type=['A',2])
sage: T is T2
True
sage: T.category()
Category of tensor products of classical crystals
sage: T3 = crystals.TensorProduct(C, C, C)
sage: T3p = crystals.TensorProduct(T, C)
sage: T3 is T3p
True
sage: B1 = crystals.TensorProduct(T, C)
sage: B2 = crystals.TensorProduct(C, T)
sage: B3 = crystals.TensorProduct(C, C, C)
sage: B1 is B2 and B2 is B3
True
sage: B = crystals.infinity.Tableaux(['A',2])
sage: T = crystals.TensorProduct(B, B)
sage: T.category()
Category of infinite tensor products of highest weight crystals
TESTS:
Check that mismatched Cartan types raise an error::
sage: A2 = crystals.Letters(['A', 2])
sage: A3 = crystals.Letters(['A', 3])
sage: crystals.TensorProduct(A2, A3)
Traceback (most recent call last):
...
ValueError: all crystals must be of the same Cartan type
"""
crystals = tuple(crystals)
if "cartan_type" in options:
cartan_type = CartanType(options.pop("cartan_type"))
else:
if not crystals:
raise ValueError(
"you need to specify the Cartan type if the tensor product list is empty"
)
else:
cartan_type = crystals[0].cartan_type()
if any(c.cartan_type() != cartan_type for c in crystals):
raise ValueError("all crystals must be of the same Cartan type")
if "generators" in options:
generators = tuple(
tuple(x) if isinstance(x, list) else x
for x in options["generators"])
if all(c in RegularCrystals() for c in crystals):
return TensorProductOfRegularCrystalsWithGenerators(
crystals, generators, cartan_type)
return TensorProductOfCrystalsWithGenerators(
crystals, generators, cartan_type)
# Flatten out tensor products
tp = sum([
B.crystals if isinstance(B, FullTensorProductOfCrystals) else (B, )
for B in crystals
], ())
if all(c in RegularCrystals() for c in crystals):
return FullTensorProductOfRegularCrystals(tp,
cartan_type=cartan_type)
return FullTensorProductOfCrystals(tp, cartan_type=cartan_type)
def __classcall_private__(cls, cartan_type, shapes=None, shape=None):
"""
Normalizes the input arguments to ensure unique representation,
and to delegate the construction of spin tableaux.
EXAMPLES::
sage: T1 = crystals.Tableaux(CartanType(['A',3]), shape = [2,2])
sage: T2 = crystals.Tableaux(['A',3], shape = (2,2))
sage: T3 = crystals.Tableaux(['A',3], shapes = ([2,2],))
sage: T2 is T1, T3 is T1
(True, True)
sage: T1 = crystals.Tableaux(['A', [1,1]], shape=[3,1,1,1])
sage: T1
Crystal of BKK tableaux of shape [3, 1, 1, 1] of gl(2|2)
sage: T2 = crystals.Tableaux(['A', [1,1]], [3,1,1,1])
sage: T1 is T2
True
"""
cartan_type = CartanType(cartan_type)
if cartan_type.letter == 'A' and isinstance(cartan_type,
SuperCartanType_standard):
if shape is None:
shape = shapes
from sage.combinat.crystals.bkk_crystals import CrystalOfBKKTableaux
return CrystalOfBKKTableaux(cartan_type, shape=shape)
n = cartan_type.rank()
# standardize shape/shapes input into a tuple of tuples
assert operator.xor(shape is not None, shapes is not None)
if shape is not None:
shapes = (shape, )
spin_shapes = tuple(tuple(shape) for shape in shapes)
try:
shapes = tuple(
tuple(trunc(i) for i in shape) for shape in spin_shapes)
except Exception:
raise ValueError(
"shapes should all be partitions or half-integer partitions")
if spin_shapes == shapes:
return super(CrystalOfTableaux,
cls).__classcall__(cls, cartan_type, shapes)
# Handle the construction of a crystals of spin tableaux
# Caveat: this currently only supports all shapes being half
# integer partitions of length the rank for type B and D. In
# particular, for type D, the spins all have to be plus or all
# minus spins
if any(len(sh) != n for sh in shapes):
raise ValueError(
"the length of all half-integer partition shapes should be the rank"
)
if any(2 * i % 2 != 1 for shape in spin_shapes for i in shape):
raise ValueError(
"shapes should be either all partitions or all half-integer partitions"
)
if cartan_type.type() == 'D':
if all(i >= 0 for shape in spin_shapes for i in shape):
S = CrystalOfSpinsPlus(cartan_type)
elif all(shape[-1] < 0 for shape in spin_shapes):
S = CrystalOfSpinsMinus(cartan_type)
else:
raise ValueError(
"in type D spins should all be positive or negative")
else:
if any(i < 0 for shape in spin_shapes for i in shape):
raise ValueError("shapes should all be partitions")
S = CrystalOfSpins(cartan_type)
B = CrystalOfTableaux(cartan_type, shapes=shapes)
T = TensorProductOfCrystals(S,
B,
generators=[[S.module_generators[0], x]
for x in B.module_generators])
T.rename("The crystal of tableaux of type %s and shape(s) %s" %
(cartan_type, list(list(shape) for shape in spin_shapes)))
T.shapes = spin_shapes
return T
def __classcall_private__(cls,
R=None,
arg0=None,
arg1=None,
names=None,
index_set=None,
abelian=False,
**kwds):
"""
Select the correct parent based upon input.
TESTS::
sage: LieAlgebra(QQ, abelian=True, names='x,y,z')
Abelian Lie algebra on 3 generators (x, y, z) over Rational Field
sage: LieAlgebra(QQ, {('e','h'): {'e':-2}, ('f','h'): {'f':2},
....: ('e','f'): {'h':1}}, names='e,f,h')
Lie algebra on 3 generators (e, f, h) over Rational Field
"""
# Parse associative algebra input
# -----
assoc = kwds.get("associative", None)
if assoc is not None:
return LieAlgebraFromAssociative(assoc,
names=names,
index_set=index_set)
# Parse input as a Cartan type
# -----
ct = kwds.get("cartan_type", None)
if ct is not None:
from sage.combinat.root_system.cartan_type import CartanType
ct = CartanType(ct)
if ct.is_affine():
from sage.algebras.lie_algebras.affine_lie_algebra import AffineLieAlgebra
return AffineLieAlgebra(R,
cartan_type=ct,
kac_moody=kwds.get("kac_moody", True))
if not ct.is_finite():
raise NotImplementedError(
"non-finite types are not implemented yet, see trac #14901 for details"
)
rep = kwds.get("representation", "bracket")
if rep == 'bracket':
from sage.algebras.lie_algebras.classical_lie_algebra import LieAlgebraChevalleyBasis
return LieAlgebraChevalleyBasis(R, ct)
if rep == 'matrix':
from sage.algebras.lie_algebras.classical_lie_algebra import ClassicalMatrixLieAlgebra
return ClassicalMatrixLieAlgebra(R, ct)
raise ValueError("invalid representation")
# Parse the remaining arguments
# -----
if R is None:
raise ValueError("invalid arguments")
check_assoc = lambda A: (isinstance(A, (Ring, MatrixSpace)) or A in
Rings() or A in Algebras(R).Associative())
if arg0 in ZZ or check_assoc(arg1):
# Check if we need to swap the arguments
arg0, arg1 = arg1, arg0
# Parse the first argument
# -----
if isinstance(arg0, dict):
if not arg0:
from sage.algebras.lie_algebras.abelian import AbelianLieAlgebra
return AbelianLieAlgebra(R, names, index_set)
elif isinstance(arg0.keys()[0], (list, tuple)):
# We assume it is some structure coefficients
arg1, arg0 = arg0, arg1
if isinstance(arg0, (list, tuple)):
if all(isinstance(x, str) for x in arg0):
# If they are all strings, then it is a list of variables
names = tuple(arg0)
if isinstance(arg0, str):
names = tuple(arg0.split(','))
elif isinstance(names, str):
names = tuple(names.split(','))
# Parse the second argument
if isinstance(arg1, dict):
# Assume it is some structure coefficients
from sage.algebras.lie_algebras.structure_coefficients import LieAlgebraWithStructureCoefficients
return LieAlgebraWithStructureCoefficients(R, arg1, names,
index_set, **kwds)
# Otherwise it must be either a free or abelian Lie algebra
if arg1 in ZZ:
if isinstance(arg0, str):
names = arg0
if names is None:
index_set = list(range(arg1))
else:
if isinstance(names, str):
names = tuple(names.split(','))
if arg1 != 1 and len(names) == 1:
names = tuple('{}{}'.format(names[0], i)
for i in range(arg1))
if arg1 != len(names):
raise ValueError("the number of names must equal the"
" number of generators")
if abelian:
from sage.algebras.lie_algebras.abelian import AbelianLieAlgebra
return AbelianLieAlgebra(R, names, index_set)
# Otherwise it is the free Lie algebra
rep = kwds.get("representation", "bracket")
if rep == "polynomial":
# Construct the free Lie algebra from polynomials in the
# free (associative unital) algebra
# TODO: Change this to accept an index set once FreeAlgebra accepts one
from sage.algebras.free_algebra import FreeAlgebra
F = FreeAlgebra(R, names)
if index_set is None:
index_set = F.variable_names()
# TODO: As part of #16823, this should instead construct a
# subclass with specialized methods for the free Lie algebra
return LieAlgebraFromAssociative(F,
F.gens(),
names=names,
index_set=index_set)
raise NotImplementedError("the free Lie algebra has only been"
" implemented using polynomials in the"
" free algebra, see trac ticket #16823")
def cartan_type(self):
r"""
Returns the Cartan type of the Cartan companion of self.b_matrix()
Only crystallographic types are implemented
Warning: this function is redundant but the corresonding method in
CartanType does not recognize all the types
"""
A = self.cartan_companion()
n = self.rk
degrees_dict = dict(zip(range(n), map(sum, 2 - A)))
degrees_set = Set(degrees_dict.values())
types_to_check = [["A", n]]
if n > 1:
types_to_check.append(["B", n])
if n > 2:
types_to_check.append(["C", n])
if n > 3:
types_to_check.append(["D", n])
if n >= 6 and n <= 8:
types_to_check.append(["E", n])
if n == 4:
types_to_check.append(["F", n])
if n == 2:
types_to_check.append(["G", n])
if n > 1:
types_to_check.append(["A", n - 1, 1])
types_to_check.append(["B", n - 1, 1])
types_to_check.append(["BC", n - 1, 2])
types_to_check.append(["A", 2 * n - 2, 2])
types_to_check.append(["A", 2 * n - 3, 2])
if n > 2:
types_to_check.append(["C", n - 1, 1])
types_to_check.append(["D", n, 2])
if n > 3:
types_to_check.append(["D", n - 1, 1])
if n >= 7 and n <= 9:
types_to_check.append(["E", n - 1, 1])
if n == 5:
types_to_check.append(["F", 4, 1])
if n == 3:
types_to_check.append(["G", n - 1, 1])
types_to_check.append(["D", 4, 3])
if n == 5:
types_to_check.append(["E", 6, 2])
for ct_name in types_to_check:
ct = CartanType(ct_name)
if 0 not in ct.index_set():
ct = ct.relabel(dict(zip(range(1, n + 1), range(n))))
ct_matrix = ct.cartan_matrix()
ct_degrees_dict = dict(zip(range(n), map(sum, 2 - ct_matrix)))
if Set(ct_degrees_dict.values()) != degrees_set:
continue
for p in Permutations(range(n)):
relabeling = dict(zip(range(n), p))
ct_new = ct.relabel(relabeling)
if ct_new.cartan_matrix() == A:
return copy(ct_new)
raise ValueError("Type not recognized")
def __classcall_private__(cls,
ambient,
contained=None,
generators=None,
virtualization=None,
scaling_factors=None,
cartan_type=None,
index_set=None,
category=None):
"""
Normalize arguments to ensure a (relatively) unique representation.
EXAMPLES::
sage: B = crystals.Tableaux(['A',4], shape=[2,1])
sage: S1 = B.subcrystal(generators=(B(2,1,1), B(5,2,4)), index_set=[1,2])
sage: S2 = B.subcrystal(generators=[B(2,1,1), B(5,2,4)], cartan_type=['A',4], index_set=(1,2))
sage: S1 is S2
True
"""
if isinstance(contained, (list, tuple, set, frozenset)):
contained = frozenset(contained)
#elif contained in Sets():
if cartan_type is None:
cartan_type = ambient.cartan_type()
else:
cartan_type = CartanType(cartan_type)
if index_set is None:
index_set = cartan_type.index_set
if generators is None:
generators = ambient.module_generators
category = Crystals().or_subcategory(category)
if ambient in FiniteCrystals() or isinstance(contained, frozenset):
category = category.Finite()
if ambient in RegularSuperCrystals():
category = category & RegularSuperCrystals()
if virtualization is not None:
if scaling_factors is None:
scaling_factors = {i: 1 for i in index_set}
from sage.combinat.crystals.virtual_crystal import VirtualCrystal
return VirtualCrystal(ambient, virtualization, scaling_factors,
contained, generators, cartan_type,
index_set, category)
if scaling_factors is not None:
# virtualization must be None
virtualization = {i: (i, ) for i in index_set}
from sage.combinat.crystals.virtual_crystal import VirtualCrystal
return VirtualCrystal(ambient, virtualization, scaling_factors,
contained, generators, cartan_type,
index_set, category)
# We need to give these as optional arguments so it unpickles correctly
return super(Subcrystal, cls).__classcall__(cls,
ambient,
contained,
tuple(generators),
cartan_type=cartan_type,
index_set=tuple(index_set),
category=category)
def Associahedron(cartan_type):
r"""
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:`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.
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
TESTS::
sage: sorted(polytopes.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(polytopes.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: polytopes.associahedron(['A',4]).f_vector()
(1, 42, 84, 56, 14, 1)
sage: polytopes.associahedron(['B',4]).f_vector()
(1, 70, 140, 90, 20, 1)
"""
cartan_type = CartanType(cartan_type)
parent = Associahedra(QQ, cartan_type.rank(), 'ppl')
return parent(cartan_type)
def FundamentalGroupOfExtendedAffineWeylGroup(cartan_type,
prefix='pi',
general_linear=None):
r"""
Factory for the fundamental group of an extended affine Weyl group.
INPUT:
- ``cartan_type`` -- a Cartan type that is either affine or finite, with the latter being a
shorthand for the untwisted affinization
- ``prefix`` (default: 'pi') -- string that labels the elements of the group
- ``general_linear`` -- (default: None, meaning False) In untwisted type A, if True, use the
universal central extension
.. RUBRIC:: Fundamental group
Associated to each affine Cartan type `\tilde{X}` is an extended affine Weyl group `E`.
Its subgroup of length-zero elements is called the fundamental group `F`.
The group `F` can be identified with a subgroup of the group of automorphisms of the
affine Dynkin diagram. As such, every element of `F` can be viewed as a permutation of the
set `I` of affine Dynkin nodes.
Let `0 \in I` be the distinguished affine node; it is the one whose removal produces the
associated finite Cartan type (call it `X`). A node `i \in I` is called *special*
if some automorphism of the affine Dynkin diagram, sends `0` to `i`.
The node `0` is always special due to the identity automorphism.
There is a bijection of the set of special nodes with the fundamental group. We denote the
image of `i` by `\pi_i`. The structure of `F` is determined as follows.
- `\tilde{X}` is untwisted -- `F` is isomorphic to `P^\vee/Q^\vee` where `P^\vee` and `Q^\vee` are the
coweight and coroot lattices of type `X`. The group `P^\vee/Q^\vee` consists of the cosets `\omega_i^\vee + Q^\vee`
for special nodes `i`, where `\omega_0^\vee = 0` by convention. In this case the special nodes `i`
are the *cominuscule* nodes, the ones such that `\omega_i^\vee(\alpha_j)` is `0` or `1` for all `j\in I_0 = I \setminus \{0\}`.
For `i` special, addition by `\omega_i^\vee+Q^\vee` permutes `P^\vee/Q^\vee` and therefore permutes the set of special nodes.
This permutation extends uniquely to an automorphism of the affine Dynkin diagram.
- `\tilde{X}` is dual untwisted -- (that is, the dual of `\tilde{X}` is untwisted) `F` is isomorphic to `P/Q`
where `P` and `Q` are the weight and root lattices of type `X`. The group `P/Q` consists of the cosets
`\omega_i + Q` for special nodes `i`, where `\omega_0 = 0` by convention. In this case the special nodes `i`
are the *minuscule* nodes, the ones such that `\alpha_j^\vee(\omega_i)` is `0` or `1` for all `j \in I_0`.
For `i` special, addition by `\omega_i+Q` permutes `P/Q` and therefore permutes the set of special nodes.
This permutation extends uniquely to an automorphism of the affine Dynkin diagram.
- `\tilde{X}` is mixed -- (that is, not of the above two types) `F` is the trivial group.
EXAMPLES::
sage: from sage.combinat.root_system.fundamental_group import FundamentalGroupOfExtendedAffineWeylGroup
sage: F = FundamentalGroupOfExtendedAffineWeylGroup(['A',3,1]); F
Fundamental group of type ['A', 3, 1]
sage: F.cartan_type().dynkin_diagram()
0
O-------+
| |
| |
O---O---O
1 2 3
A3~
sage: F.special_nodes()
(0, 1, 2, 3)
sage: F(1)^2
pi[2]
sage: F(1)*F(2)
pi[3]
sage: F(3)^(-1)
pi[1]
sage: F = FundamentalGroupOfExtendedAffineWeylGroup("B3"); F
Fundamental group of type ['B', 3, 1]
sage: F.cartan_type().dynkin_diagram()
O 0
|
|
O---O=>=O
1 2 3
B3~
sage: F.special_nodes()
(0, 1)
sage: F = FundamentalGroupOfExtendedAffineWeylGroup("C2"); F
Fundamental group of type ['C', 2, 1]
sage: F.cartan_type().dynkin_diagram()
O=>=O=<=O
0 1 2
C2~
sage: F.special_nodes()
(0, 2)
sage: F = FundamentalGroupOfExtendedAffineWeylGroup("D4"); F
Fundamental group of type ['D', 4, 1]
sage: F.cartan_type().dynkin_diagram()
O 4
|
|
O---O---O
1 |2 3
|
O 0
D4~
sage: F.special_nodes()
(0, 1, 3, 4)
sage: (F(4), F(4)^2)
(pi[4], pi[0])
sage: F = FundamentalGroupOfExtendedAffineWeylGroup("D5"); F
Fundamental group of type ['D', 5, 1]
sage: F.cartan_type().dynkin_diagram()
0 O O 5
| |
| |
O---O---O---O
1 2 3 4
D5~
sage: F.special_nodes()
(0, 1, 4, 5)
sage: (F(5), F(5)^2, F(5)^3, F(5)^4)
(pi[5], pi[1], pi[4], pi[0])
sage: F = FundamentalGroupOfExtendedAffineWeylGroup("E6"); F
Fundamental group of type ['E', 6, 1]
sage: F.cartan_type().dynkin_diagram()
O 0
|
|
O 2
|
|
O---O---O---O---O
1 3 4 5 6
E6~
sage: F.special_nodes()
(0, 1, 6)
sage: F(1)^2
pi[6]
sage: F = FundamentalGroupOfExtendedAffineWeylGroup(['D',4,2]); F
Fundamental group of type ['C', 3, 1]^*
sage: F.cartan_type().dynkin_diagram()
O=<=O---O=>=O
0 1 2 3
C3~*
sage: F.special_nodes()
(0, 3)
We also implement a fundamental group for `GL_n`. It is defined to be the group of integers, which is the
covering group of the fundamental group Z/nZ for affine `SL_n`::
sage: F = FundamentalGroupOfExtendedAffineWeylGroup(['A',2,1], general_linear=True); F
Fundamental group of GL(3)
sage: x = F.an_element(); x
pi[5]
sage: x*x
pi[10]
sage: x.inverse()
pi[-5]
sage: wt = F.cartan_type().classical().root_system().ambient_space().an_element(); wt
(2, 2, 3)
sage: x.act_on_classical_ambient(wt)
(2, 3, 2)
sage: w = WeylGroup(F.cartan_type(),prefix="s").an_element(); w
s0*s1*s2
sage: x.act_on_affine_weyl(w)
s2*s0*s1
"""
cartan_type = CartanType(cartan_type)
if cartan_type.is_finite():
cartan_type = cartan_type.affine()
if not cartan_type.is_affine():
raise NotImplementedError("Cartan type is not affine")
if general_linear is True:
if cartan_type.is_untwisted_affine() and cartan_type.type() == "A":
return FundamentalGroupGL(cartan_type, prefix)
else:
raise ValueError(
"General Linear Fundamental group is untwisted type A")
return FundamentalGroupOfExtendedAffineWeylGroup_Class(cartan_type,
prefix,
finite=True)
def __classcall_private__(cls, *args, **kwds):
"""
Normalize input so we can inherit from spare integer matrix.
.. NOTE::
To disable the Cartan type check, use the optional argument
``cartan_type_check = False``.
EXAMPLES::
sage: C = CartanMatrix(['A',1,1])
sage: C2 = CartanMatrix([[2, -2], [-2, 2]])
sage: C3 = CartanMatrix(matrix([[2, -2], [-2, 2]]), [0, 1])
sage: C == C2 and C == C3
True
TESTS:
Check that :trac:`15740` is fixed::
sage: d = DynkinDiagram()
sage: d.add_edge('a', 'b', 2)
sage: d.index_set()
('a', 'b')
sage: cm = CartanMatrix(d)
sage: cm.index_set()
('a', 'b')
"""
# Special case with 0 args and kwds has Cartan type
if "cartan_type" in kwds and len(args) == 0:
args = (CartanType(kwds["cartan_type"]), )
if len(args) == 0:
data = []
n = 0
index_set = tuple()
cartan_type = None
subdivisions = None
elif len(args) == 4 and isinstance(args[0],
MatrixSpace): # For pickling
return typecall(cls, args[0], args[1], args[2], args[3])
elif isinstance(args[0], CartanMatrix):
return args[0]
else:
cartan_type = None
dynkin_diagram = None
subdivisions = None
from sage.combinat.root_system.dynkin_diagram import DynkinDiagram_class
if isinstance(args[0], DynkinDiagram_class):
dynkin_diagram = args[0]
cartan_type = args[0]._cartan_type
else:
try:
cartan_type = CartanType(args[0])
dynkin_diagram = cartan_type.dynkin_diagram()
except (TypeError, ValueError):
pass
if dynkin_diagram is not None:
n = dynkin_diagram.rank()
index_set = dynkin_diagram.index_set()
reverse = dict(
(index_set[i], i) for i in range(len(index_set)))
data = {(i, i): 2 for i in range(n)}
for (i, j, l) in dynkin_diagram.edge_iterator():
data[(reverse[j], reverse[i])] = -l
else:
M = matrix(args[0])
if not is_generalized_cartan_matrix(M):
raise ValueError(
"the input matrix is not a generalized Cartan matrix")
n = M.ncols()
if "cartan_type" in kwds:
cartan_type = CartanType(kwds["cartan_type"])
elif n == 1:
cartan_type = CartanType(['A', 1])
elif kwds.get("cartan_type_check", True):
cartan_type = find_cartan_type_from_matrix(M)
data = M.dict()
subdivisions = M._subdivisions
if len(args) == 1:
if cartan_type is not None:
index_set = tuple(cartan_type.index_set())
elif dynkin_diagram is None:
index_set = tuple(range(n))
elif len(args) == 2:
index_set = tuple(args[1])
if len(index_set) != n and len(set(index_set)) != n:
raise ValueError("the given index set is not valid")
else:
raise ValueError("too many arguments")
mat = typecall(cls, MatrixSpace(ZZ, n, sparse=True), data, cartan_type,
index_set)
mat._subdivisions = subdivisions
return mat
def CoxeterGroup(data, implementation=None, base_ring=None, index_set=None):
"""
Return an implementation of the Coxeter group given by ``data``.
INPUT:
- ``data`` -- a cartan type (or coercible into; see :class:`CartanType`)
or a Coxeter matrix or graph
- ``implementation`` -- (default: ``None``) can be one of the following:
* "permutation" - as a permutation representation
* "matrix" - as a Weyl group (as a matrix group acting on the root space);
if this is not implemented, this uses the "reflection" implementation
* "coxeter3" - using the coxeter3 package
* "reflection" - as elements in the reflection representation; see
:class:`~sage.groups.matrix_gps.coxeter_groups.CoxeterMatrixGroup`
* ``None`` - choose "permutation" representation if possible, else
default to "matrix"
- ``base_ring`` -- (optional) the base ring for the "reflection"
implementation
- ``index_set`` -- (optional) the index set for the "reflection"
implementation
EXAMPLES:
Now assume that ``data`` represents a Cartan type. If
``implementation`` is not specified, a permutation
representation is returned whenever possible (finite irreducible
Cartan type, with the GAP3 Chevie package available)::
sage: W = CoxeterGroup(["A",2])
sage: W # optional - chevie
Permutation Group with generators [(1,3)(2,5)(4,6), (1,4)(2,3)(5,6)]
Otherwise, a matrix representation is returned::
sage: W = CoxeterGroup(["A",3,1])
sage: W
Weyl Group of type ['A', 3, 1] (as a matrix group acting on the root space)
sage: W = CoxeterGroup(['H',3])
sage: W
Coxeter group over Universal Cyclotomic Field with Coxeter matrix:
[1 3 2]
[3 1 5]
[2 5 1]
We now use the ``implementation`` option::
sage: W = CoxeterGroup(["A",2], implementation = "permutation") # optional - chevie
sage: W # optional - chevie
Permutation Group with generators [(1,3)(2,5)(4,6), (1,4)(2,3)(5,6)]
sage: W.category() # optional - chevie
Join of Category of finite permutation groups and Category of finite coxeter groups
sage: W = CoxeterGroup(["A",2], implementation = "matrix")
sage: W
Weyl Group of type ['A', 2] (as a matrix group acting on the ambient space)
sage: W = CoxeterGroup(["H",3], implementation = "matrix")
sage: W
Coxeter group over Universal Cyclotomic Field with Coxeter matrix:
[1 3 2]
[3 1 5]
[2 5 1]
sage: W = CoxeterGroup(["H",3], implementation="reflection")
sage: W
Coxeter group over Universal Cyclotomic Field with Coxeter matrix:
[1 3 2]
[3 1 5]
[2 5 1]
sage: W = CoxeterGroup(["A",4,1], implementation = "permutation")
Traceback (most recent call last):
...
NotImplementedError: Coxeter group of type ['A', 4, 1] as permutation group not implemented
We use the different options for the "reflection" implementation::
sage: W = CoxeterGroup(["H",3], implementation="reflection", base_ring=RR)
sage: W
Coxeter group over Real Field with 53 bits of precision with Coxeter matrix:
[1 3 2]
[3 1 5]
[2 5 1]
sage: W = CoxeterGroup([[1,10],[10,1]], implementation="reflection", index_set=['a','b'], base_ring=SR)
sage: W
Coxeter group over Symbolic Ring with Coxeter matrix:
[ 1 10]
[10 1]
"""
if implementation not in [
"permutation", "matrix", "coxeter3", "reflection", None
]:
raise ValueError("invalid type implementation")
try:
cartan_type = CartanType(data)
except (
TypeError, ValueError
): # If it is not a Cartan type, try to see if we can represent it as a matrix group
return CoxeterMatrixGroup(data, base_ring, index_set)
if implementation is None:
if cartan_type.is_finite() and cartan_type.is_irreducible(
) and is_chevie_available():
implementation = "permutation"
else:
implementation = "matrix"
if implementation == "reflection":
return CoxeterMatrixGroup(cartan_type, base_ring, index_set)
if implementation == "coxeter3":
try:
from sage.libs.coxeter3.coxeter_group import CoxeterGroup
except ImportError:
raise RuntimeError("coxeter3 must be installed")
else:
return CoxeterGroup(cartan_type)
if implementation == "permutation" and is_chevie_available() and \
cartan_type.is_finite() and cartan_type.is_irreducible():
return CoxeterGroupAsPermutationGroup(cartan_type)
elif implementation == "matrix":
if cartan_type.is_crystallographic():
return WeylGroup(cartan_type)
return CoxeterMatrixGroup(cartan_type, base_ring, index_set)
raise NotImplementedError(
"Coxeter group of type %s as %s group not implemented " %
(cartan_type, implementation))
def WeylGroup(x, prefix=None, implementation='matrix'):
"""
Returns the Weyl group of the root system defined by the Cartan
type (or matrix) ``ct``.
INPUT:
- ``x`` - a root system or a Cartan type (or matrix)
OPTIONAL:
- ``prefix`` -- changes the representation of elements from matrices
to products of simple reflections
- ``implementation`` -- one of the following:
* ``'matrix'`` - as matrices acting on a root system
* ``"permutation"`` - as a permutation group acting on the roots
EXAMPLES:
The following constructions yield the same result, namely
a weight lattice and its corresponding Weyl group::
sage: G = WeylGroup(['F',4])
sage: L = G.domain()
or alternatively and equivalently::
sage: L = RootSystem(['F',4]).ambient_space()
sage: G = L.weyl_group()
sage: W = WeylGroup(L)
Either produces a weight lattice, with access to its roots and
weights.
::
sage: G = WeylGroup(['F',4])
sage: G.order()
1152
sage: [s1,s2,s3,s4] = G.simple_reflections()
sage: w = s1*s2*s3*s4; w
[ 1/2 1/2 1/2 1/2]
[-1/2 1/2 1/2 -1/2]
[ 1/2 1/2 -1/2 -1/2]
[ 1/2 -1/2 1/2 -1/2]
sage: type(w) == G.element_class
True
sage: w.order()
12
sage: w.length() # length function on Weyl group
4
The default representation of Weyl group elements is as matrices.
If you prefer, you may specify a prefix, in which case the
elements are represented as products of simple reflections.
::
sage: W=WeylGroup("C3",prefix="s")
sage: [s1,s2,s3]=W.simple_reflections() # lets Sage parse its own output
sage: s2*s1*s2*s3
s1*s2*s3*s1
sage: s2*s1*s2*s3 == s1*s2*s3*s1
True
sage: (s2*s3)^2==(s3*s2)^2
True
sage: (s1*s2*s3*s1).matrix()
[ 0 0 -1]
[ 0 1 0]
[ 1 0 0]
::
sage: L = G.domain()
sage: fw = L.fundamental_weights(); fw
Finite family {1: (1, 1, 0, 0), 2: (2, 1, 1, 0), 3: (3/2, 1/2, 1/2, 1/2), 4: (1, 0, 0, 0)}
sage: rho = sum(fw); rho
(11/2, 5/2, 3/2, 1/2)
sage: w.action(rho) # action of G on weight lattice
(5, -1, 3, 2)
We can also do the same for arbitrary Cartan matrices::
sage: cm = CartanMatrix([[2,-5,0],[-2,2,-1],[0,-1,2]])
sage: W = WeylGroup(cm)
sage: W.gens()
(
[-1 5 0] [ 1 0 0] [ 1 0 0]
[ 0 1 0] [ 2 -1 1] [ 0 1 0]
[ 0 0 1], [ 0 0 1], [ 0 1 -1]
)
sage: s0,s1,s2 = W.gens()
sage: s1*s2*s1
[ 1 0 0]
[ 2 0 -1]
[ 2 -1 0]
sage: s2*s1*s2
[ 1 0 0]
[ 2 0 -1]
[ 2 -1 0]
sage: s0*s1*s0*s2*s0
[ 9 0 -5]
[ 2 0 -1]
[ 0 1 -1]
Same Cartan matrix, but with a prefix to display using simple reflections::
sage: W = WeylGroup(cm, prefix='s')
sage: s0,s1,s2 = W.gens()
sage: s0*s2*s1
s2*s0*s1
sage: (s1*s2)^3
1
sage: (s0*s1)^5
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1
sage: s0*s1*s2*s1*s2
s2*s0*s1
sage: s0*s1*s2*s0*s2
s0*s1*s0
TESTS::
sage: TestSuite(WeylGroup(["A",3])).run()
sage: TestSuite(WeylGroup(["A",3,1])).run() # long time
sage: W = WeylGroup(['A',3,1])
sage: s = W.simple_reflections()
sage: w = s[0]*s[1]*s[2]
sage: w.reduced_word()
[0, 1, 2]
sage: w = s[0]*s[2]
sage: w.reduced_word()
[2, 0]
sage: W = groups.misc.WeylGroup(['A',3,1])
"""
if implementation == "permutation":
return WeylGroup_permutation(x, prefix)
elif implementation != "matrix":
raise ValueError("invalid implementation")
if x in RootLatticeRealizations:
return WeylGroup_gens(x, prefix=prefix)
try:
ct = CartanType(x)
except TypeError:
ct = CartanMatrix(x) # See if it is a Cartan matrix
if ct.is_finite():
return WeylGroup_gens(ct.root_system().ambient_space(), prefix=prefix)
return WeylGroup_gens(ct.root_system().root_space(), prefix=prefix)
def WeylGroup(x, prefix=None):
"""
Returns the Weyl group of type ct.
INPUT:
- ``ct`` - a Cartan Type.
OPTIONAL:
- ``prefix`` - changes the representation of elements from matrices
to products of simple reflections
EXAMPLES: The following constructions yield the same result, namely
a weight lattice and its corresponding Weyl group::
sage: G = WeylGroup(['F',4])
sage: L = G.domain()
or alternatively and equivalently::
sage: L = RootSystem(['F',4]).ambient_space()
sage: G = L.weyl_group()
Either produces a weight lattice, with access to its roots and
weights.
::
sage: G = WeylGroup(['F',4])
sage: G.order()
1152
sage: [s1,s2,s3,s4] = G.simple_reflections()
sage: w = s1*s2*s3*s4; w
[ 1/2 1/2 1/2 1/2]
[-1/2 1/2 1/2 -1/2]
[ 1/2 1/2 -1/2 -1/2]
[ 1/2 -1/2 1/2 -1/2]
sage: type(w) == G.element_class
True
sage: w.order()
12
sage: w.length() # length function on Weyl group
4
The default representation of Weyl group elements is as matrices.
If you prefer, you may specify a prefix, in which case the
elements are represented as products of simple reflections.
::
sage: W=WeylGroup("C3",prefix="s")
sage: [s1,s2,s3]=W.simple_reflections() # lets Sage parse its own output
sage: s2*s1*s2*s3
s1*s2*s3*s1
sage: s2*s1*s2*s3 == s1*s2*s3*s1
True
sage: (s2*s3)^2==(s3*s2)^2
True
sage: (s1*s2*s3*s1).matrix()
[ 0 0 -1]
[ 0 1 0]
[ 1 0 0]
::
sage: L = G.domain()
sage: fw = L.fundamental_weights(); fw
Finite family {1: (1, 1, 0, 0), 2: (2, 1, 1, 0), 3: (3/2, 1/2, 1/2, 1/2), 4: (1, 0, 0, 0)}
sage: rho = sum(fw); rho
(11/2, 5/2, 3/2, 1/2)
sage: w.action(rho) # action of G on weight lattice
(5, -1, 3, 2)
TESTS::
sage: TestSuite(WeylGroup(["A",3])).run()
sage: TestSuite(WeylGroup(["A",3, 1])).run()
sage: W=WeylGroup(['A',3,1])
sage: s=W.simple_reflections()
sage: w=s[0]*s[1]*s[2]
sage: w.reduced_word()
[0, 1, 2]
sage: w=s[0]*s[2]
sage: w.reduced_word()
[2, 0]
"""
if x in RootLatticeRealizations:
return WeylGroup_gens(x, prefix=prefix)
ct = CartanType(x)
if ct.is_affine():
return WeylGroup_gens(ct.root_system().root_space(), prefix=prefix)
else:
return WeylGroup_gens(ct.root_system().ambient_space(), prefix=prefix)
def CoxeterGroup(data,
implementation="reflection",
base_ring=None,
index_set=None):
"""
Return an implementation of the Coxeter group given by ``data``.
INPUT:
- ``data`` -- a Cartan type (or coercible into; see :class:`CartanType`)
or a Coxeter matrix or graph
- ``implementation`` -- (default: ``'reflection'``) can be one of
the following:
* ``'permutation'`` - as a permutation representation
* ``'matrix'`` - as a Weyl group (as a matrix group acting on the
root space); if this is not implemented, this uses the "reflection"
implementation
* ``'coxeter3'`` - using the coxeter3 package
* ``'reflection'`` - as elements in the reflection representation; see
:class:`~sage.groups.matrix_gps.coxeter_groups.CoxeterMatrixGroup`
- ``base_ring`` -- (optional) the base ring for the ``'reflection'``
implementation
- ``index_set`` -- (optional) the index set for the ``'reflection'``
implementation
EXAMPLES:
Now assume that ``data`` represents a Cartan type. If
``implementation`` is not specified, the reflection representation
is returned::
sage: W = CoxeterGroup(["A",2])
sage: W
Finite Coxeter group over Universal Cyclotomic Field with Coxeter matrix:
[1 3]
[3 1]
sage: W = CoxeterGroup(["A",3,1]); W
Coxeter group over Universal Cyclotomic Field with Coxeter matrix:
[1 3 2 3]
[3 1 3 2]
[2 3 1 3]
[3 2 3 1]
sage: W = CoxeterGroup(['H',3]); W
Finite Coxeter group over Universal Cyclotomic Field with Coxeter matrix:
[1 3 2]
[3 1 5]
[2 5 1]
We now use the ``implementation`` option::
sage: W = CoxeterGroup(["A",2], implementation = "permutation") # optional - gap3
sage: W # optional - gap3
Permutation Group with generators [(1,3)(2,5)(4,6), (1,4)(2,3)(5,6)]
sage: W.category() # optional - gap3
Join of Category of finite permutation groups
and Category of finite coxeter groups
and Category of well generated finite irreducible complex reflection groups
sage: W = CoxeterGroup(["A",2], implementation="matrix")
sage: W
Weyl Group of type ['A', 2] (as a matrix group acting on the ambient space)
sage: W = CoxeterGroup(["H",3], implementation="matrix")
sage: W
Finite Coxeter group over Universal Cyclotomic Field with Coxeter matrix:
[1 3 2]
[3 1 5]
[2 5 1]
sage: W = CoxeterGroup(["H",3], implementation="reflection")
sage: W
Finite Coxeter group over Universal Cyclotomic Field with Coxeter matrix:
[1 3 2]
[3 1 5]
[2 5 1]
sage: W = CoxeterGroup(["A",4,1], implementation="permutation")
Traceback (most recent call last):
...
NotImplementedError: Coxeter group of type ['A', 4, 1] as permutation group not implemented
sage: W = CoxeterGroup(["A",4], implementation="chevie"); W # optional - gap3
Irreducible real reflection group of rank 4 and type A4
We use the different options for the "reflection" implementation::
sage: W = CoxeterGroup(["H",3], implementation="reflection", base_ring=RR)
sage: W
Finite Coxeter group over Real Field with 53 bits of precision with Coxeter matrix:
[1 3 2]
[3 1 5]
[2 5 1]
sage: W = CoxeterGroup([[1,10],[10,1]], implementation="reflection", index_set=['a','b'], base_ring=SR)
sage: W
Finite Coxeter group over Symbolic Ring with Coxeter matrix:
[ 1 10]
[10 1]
TESTS::
sage: W = groups.misc.CoxeterGroup(["H",3])
"""
if implementation not in [
"permutation", "matrix", "coxeter3", "reflection", "chevie", None
]:
raise ValueError("invalid type implementation")
try:
cartan_type = CartanType(data)
except (
TypeError, ValueError
): # If it is not a Cartan type, try to see if we can represent it as a matrix group
return CoxeterMatrixGroup(data, base_ring, index_set)
if implementation is None:
implementation = "matrix"
if implementation == "reflection":
return CoxeterMatrixGroup(cartan_type, base_ring, index_set)
if implementation == "coxeter3":
try:
from sage.libs.coxeter3.coxeter_group import CoxeterGroup
except ImportError:
raise RuntimeError("coxeter3 must be installed")
else:
return CoxeterGroup(cartan_type)
if implementation == "permutation" and is_chevie_available() and \
cartan_type.is_finite() and cartan_type.is_irreducible():
return CoxeterGroupAsPermutationGroup(cartan_type)
elif implementation == "matrix":
if cartan_type.is_crystallographic():
return WeylGroup(cartan_type)
return CoxeterMatrixGroup(cartan_type, base_ring, index_set)
elif implementation == "chevie":
from sage.combinat.root_system.reflection_group_real import ReflectionGroup
return ReflectionGroup(data, index_set=index_set)
raise NotImplementedError(
"Coxeter group of type {} as {} group not implemented".format(
cartan_type, implementation))
def __classcall_private__(cls, data, base_ring=None, index_set=None):
"""
Normalize arguments to ensure a unique representation.
EXAMPLES::
sage: W1 = CoxeterGroup(['A',2], implementation="reflection", base_ring=UniversalCyclotomicField())
sage: W2 = CoxeterGroup([[1,3],[3,1]], index_set=(1,2))
sage: W1 is W2
True
sage: G1 = Graph([(1,2)])
sage: W3 = CoxeterGroup(G1)
sage: W1 is W3
True
sage: G2 = Graph([(1,2,3)])
sage: W4 = CoxeterGroup(G2)
sage: W1 is W4
True
Check with `\infty` because of the hack of using `-1` to represent
`\infty` in the Coxeter matrix::
sage: G = Graph([(0, 1, 3), (1, 2, oo)])
sage: W1 = CoxeterGroup(matrix([[1, 3, 2], [3,1,-1], [2,-1,1]]))
sage: W2 = CoxeterGroup(G)
sage: W1 is W2
True
sage: CoxeterGroup(W1.coxeter_graph()) is W1
True
"""
if isinstance(data, CartanType_abstract):
if index_set is None:
index_set = data.index_set()
data = data.coxeter_matrix()
elif isinstance(data, Graph):
G = data
n = G.num_verts()
# Setup the basis matrix as all 2 except 1 on the diagonal
data = matrix(ZZ, [[2] * n] * n)
for i in range(n):
data[i, i] = ZZ.one()
verts = G.vertices()
for e in G.edges():
m = e[2]
if m is None:
m = 3
elif m == infinity or m == -1: # FIXME: Hack because there is no ZZ\cup\{\infty\}
m = -1
elif m <= 1:
raise ValueError("invalid Coxeter graph label")
i = verts.index(e[0])
j = verts.index(e[1])
data[j, i] = data[i, j] = m
if index_set is None:
index_set = G.vertices()
else:
try:
data = matrix(data)
except (ValueError, TypeError):
data = CartanType(data).coxeter_matrix()
if not data.is_symmetric():
raise ValueError("the Coxeter matrix is not symmetric")
if any(d != 1 for d in data.diagonal()):
raise ValueError("the Coxeter matrix diagonal is not all 1")
if any(val <= 1 and val != -1 for i, row in enumerate(data.rows())
for val in row[i + 1:]):
raise ValueError("invalid Coxeter label")
if index_set is None:
index_set = range(data.nrows())
if base_ring is None:
base_ring = UniversalCyclotomicField()
data.set_immutable()
return super(CoxeterMatrixGroup,
cls).__classcall__(cls, data, base_ring, tuple(index_set))