class CartanTypeFindStat(object):
"""
A Cartan type class for FindStat
"""
def __init__(self, ct):
self._cartan_type = CartanType(ct)
def __hash__(self):
return hash(self._cartan_type)
def _latex_(self):
return self._cartan_type.dynkin_diagram()._latex_()
def __repr__(self):
return self._cartan_type._repr_()
_repr_ = __repr__
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):
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 __classcall_private__(cls,
data=None,
index_set=None,
cartan_type=None,
cartan_type_check=True):
"""
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)}
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 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, *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
"""
# 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
try:
cartan_type = CartanType(args[0])
dynkin_diagram = cartan_type.dynkin_diagram()
except (TypeError, ValueError):
pass
if dynkin_diagram is not None:
n = cartan_type.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())
else:
index_set = tuple(range(M.ncols()))
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 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
One can also input an indexing set by passing a tuple using the optional
argument ``index_set``.
The edge multiplicities are encoded as edge labels. For the corresponding
Cartan matrices, this uses the convention in Hong and Kang, Kac,
Fulton and 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 __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 __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 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)
