def dynkin_diagram(self):
"""
Return the Dynkin diagram corresponding to ``self``.
EXAMPLES::
sage: C = CartanMatrix(['A',2])
sage: C.dynkin_diagram()
O---O
1 2
A2
sage: C = CartanMatrix(['F',4,1])
sage: C.dynkin_diagram()
O---O---O=>=O---O
0 1 2 3 4
F4~
sage: C = CartanMatrix([[2,-4],[-4,2]])
sage: C.dynkin_diagram()
Dynkin diagram of rank 2
"""
from sage.combinat.root_system.dynkin_diagram import DynkinDiagram
if self._cartan_type is not None:
return DynkinDiagram(self._cartan_type)
from dynkin_diagram import DynkinDiagram_class
n = self.nrows()
g = DynkinDiagram_class(self)
for i in range(n):
for j in range(n):
if self[i,j] == -1:
g.add_edge(i, j)
elif self[i,j] < -1:
g.add_edge(i, j, -self[i,j])
return g
def dynkin_diagram(self):
"""
Returns a Dynkin diagram for type B.
EXAMPLES::
sage: b = CartanType(['B',3]).dynkin_diagram()
sage: b
O---O=>=O
1 2 3
B3
sage: sorted(b.edges())
[(1, 2, 1), (2, 1, 1), (2, 3, 2), (3, 2, 1)]
sage: b = CartanType(['B',1]).dynkin_diagram()
sage: b
O
1
B1
sage: sorted(b.edges())
[]
"""
from dynkin_diagram import DynkinDiagram_class
n = self.n
g = DynkinDiagram_class(self)
for i in range(1, n):
g.add_edge(i, i + 1)
if n >= 2:
g.set_edge_label(n - 1, n, 2)
return g
def dynkin_diagram(self):
"""
Returns the Dynkin diagram of type A.
EXAMPLES::
sage: a = CartanType(['A',3]).dynkin_diagram()
sage: a
O---O---O
1 2 3
A3
sage: sorted(a.edges())
[(1, 2, 1), (2, 1, 1), (2, 3, 1), (3, 2, 1)]
TEST::
sage: a = DynkinDiagram(['A',1])
sage: a
O
1
A1
sage: a.vertices(), a.edges()
([1], [])
"""
from dynkin_diagram import DynkinDiagram_class
n = self.n
g = DynkinDiagram_class(self)
for i in range(1, n):
g.add_edge(i, i + 1)
return g
def dynkin_diagram(self):
"""
Returns a Dynkin diagram for type reducible.
EXAMPLES::
sage: dd = CartanType("A2xB2xF4").dynkin_diagram()
sage: dd
O---O
1 2
O=>=O
3 4
O---O=>=O---O
5 6 7 8
A2xB2xF4
sage: dd.edges()
[(1, 2, 1), (2, 1, 1), (3, 4, 2), (4, 3, 1), (5, 6, 1), (6, 5, 1), (6, 7, 2), (7, 6, 1), (7, 8, 1), (8, 7, 1)]
sage: CartanType("F4xA2").dynkin_diagram()
O---O=>=O---O
1 2 3 4
O---O
5 6
F4xA2
"""
from dynkin_diagram import DynkinDiagram_class
relabelling = self._index_relabelling
g = DynkinDiagram_class(self)
for i in range(len(self._types)):
for [e1, e2, l] in self._types[i].dynkin_diagram().edges():
g.add_edge(relabelling[i,e1], relabelling[i,e2], label=l)
return g
def dynkin_diagram(t):
"""
Returns a Dynkin diagram for type reducible.
EXAMPLES::
sage: dd = CartanType("A2xB2xF4").dynkin_diagram()
sage: dd
O---O
1 2
O=>=O
3 4
O---O=>=O---O
5 6 7 8
A2xB2xF4
sage: dd.edges()
[(1, 2, 1), (2, 1, 1), (3, 4, 2), (4, 3, 1), (5, 6, 1), (6, 5, 1), (6, 7, 2), (7, 6, 1), (7, 8, 1), (8, 7, 1)]
sage: CartanType("F4xA2").dynkin_diagram()
O---O=>=O---O
1 2 3 4
O---O
5 6
F4xA2
"""
from dynkin_diagram import DynkinDiagram, DynkinDiagram_class
g = DynkinDiagram_class(t)
for i in range(len(t._types)):
for [e1, e2, l] in DynkinDiagram(t._types[i]).edges():
shift = t._rshifts[i]
g.add_edge(e1+shift, e2+shift, label=l)
return g
def dynkin_diagram(t):
"""
Returns a Dynkin diagram for type reducible.
EXAMPLES::
sage: dd = CartanType("A2xB2xF4").dynkin_diagram()
sage: dd
O---O
1 2
O=>=O
3 4
O---O=>=O---O
5 6 7 8
A2xB2xF4
sage: dd.edges()
[(1, 2, 1), (2, 1, 1), (3, 4, 2), (4, 3, 1), (5, 6, 1), (6, 5, 1), (6, 7, 2), (7, 6, 1), (7, 8, 1), (8, 7, 1)]
sage: CartanType("F4xA2").dynkin_diagram()
O---O=>=O---O
1 2 3 4
O---O
5 6
F4xA2
"""
from dynkin_diagram import DynkinDiagram, DynkinDiagram_class
g = DynkinDiagram_class(t)
for i in range(len(t._types)):
for [e1, e2, l] in DynkinDiagram(t._types[i]).edges():
shift = t._rshifts[i]
g.add_edge(e1 + shift, e2 + shift, label=l)
return g
def dynkin_diagram(self):
"""
Returns the Dynkin diagram of type A.
EXAMPLES::
sage: a = CartanType(['A',3]).dynkin_diagram()
sage: a
O---O---O
1 2 3
A3
sage: sorted(a.edges())
[(1, 2, 1), (2, 1, 1), (2, 3, 1), (3, 2, 1)]
TEST::
sage: a = DynkinDiagram(['A',1])
sage: a
O
1
A1
sage: a.vertices(), a.edges()
([1], [])
"""
from dynkin_diagram import DynkinDiagram_class
n = self.n
g = DynkinDiagram_class(self)
for i in range(1, n):
g.add_edge(i, i+1)
return g
def dynkin_diagram(self):
"""
Returns a Dynkin diagram for type reducible.
EXAMPLES::
sage: dd = CartanType("A2xB2xF4").dynkin_diagram()
sage: dd
O---O
1 2
O=>=O
3 4
O---O=>=O---O
5 6 7 8
A2xB2xF4
sage: dd.edges()
[(1, 2, 1), (2, 1, 1), (3, 4, 2), (4, 3, 1), (5, 6, 1), (6, 5, 1), (6, 7, 2), (7, 6, 1), (7, 8, 1), (8, 7, 1)]
sage: CartanType("F4xA2").dynkin_diagram()
O---O=>=O---O
1 2 3 4
O---O
5 6
F4xA2
"""
from dynkin_diagram import DynkinDiagram_class
relabelling = self._index_relabelling
g = DynkinDiagram_class(self)
for i in range(len(self._types)):
for [e1, e2, l] in self._types[i].dynkin_diagram().edges():
g.add_edge(relabelling[i, e1], relabelling[i, e2], label=l)
return g
def dynkin_diagram(self):
"""
Returns a Dynkin diagram for type D.
EXAMPLES::
sage: d = CartanType(['D',5]).dynkin_diagram(); d
O 5
|
|
O---O---O---O
1 2 3 4
D5
sage: sorted(d.edges())
[(1, 2, 1), (2, 1, 1), (2, 3, 1), (3, 2, 1), (3, 4, 1), (3, 5, 1), (4, 3, 1), (5, 3, 1)]
sage: d = CartanType(['D',4]).dynkin_diagram(); d
O 4
|
|
O---O---O
1 2 3
D4
sage: sorted(d.edges())
[(1, 2, 1), (2, 1, 1), (2, 3, 1), (2, 4, 1), (3, 2, 1), (4, 2, 1)]
sage: d = CartanType(['D',3]).dynkin_diagram(); d
O 3
|
|
O---O
1 2
D3
sage: sorted(d.edges())
[(1, 2, 1), (1, 3, 1), (2, 1, 1), (3, 1, 1)]
sage: d = CartanType(['D',2]).dynkin_diagram(); d
O O
1 2
D2
sage: sorted(d.edges())
[]
"""
from dynkin_diagram import DynkinDiagram_class
g = DynkinDiagram_class(self)
n = self.n
if n >= 3:
for i in range(1, n-1):
g.add_edge(i, i+1)
g.add_edge(n-2, n)
return g
def dynkin_diagram(self):
"""
Returns the extended Dynkin diagram for affine type BC.
EXAMPLES::
sage: c = CartanType(['BC',3,2]).dynkin_diagram()
sage: c
O=<=O---O=<=O
0 1 2 3
BC3~
sage: sorted(c.edges())
[(0, 1, 1), (1, 0, 2), (1, 2, 1), (2, 1, 1), (2, 3, 1), (3, 2, 2)]
sage: c = CartanType(["A", 6, 2]).dynkin_diagram() # should be the same as above; did fail at some point!
sage: c
O=<=O---O=<=O
0 1 2 3
BC3~
sage: sorted(c.edges())
[(0, 1, 1), (1, 0, 2), (1, 2, 1), (2, 1, 1), (2, 3, 1), (3, 2, 2)]
sage: c = CartanType(['BC',2,2]).dynkin_diagram()
sage: c
O=<=O=<=O
0 1 2
BC2~
sage: sorted(c.edges())
[(0, 1, 1), (1, 0, 2), (1, 2, 1), (2, 1, 2)]
sage: c = CartanType(['BC',1,2]).dynkin_diagram()
sage: c
4
O=<=O
0 1
BC1~
sage: sorted(c.edges())
[(0, 1, 1), (1, 0, 4)]
"""
from dynkin_diagram import DynkinDiagram_class
n = self.n
g = DynkinDiagram_class(self)
if n == 1:
g.add_edge(1, 0, 4)
return g
g.add_edge(1, 0, 2)
for i in range(1, n - 1):
g.add_edge(i, i + 1)
g.add_edge(n, n - 1, 2)
return g
def dynkin_diagram(self):
"""
Returns the extended Dynkin diagram for affine type C.
EXAMPLES::
sage: c = CartanType(['C',3,1]).dynkin_diagram()
sage: c
O=>=O---O=<=O
0 1 2 3
C3~
sage: sorted(c.edges())
[(0, 1, 2), (1, 0, 1), (1, 2, 1), (2, 1, 1), (2, 3, 1), (3, 2, 2)]
"""
n = self.n
if n == 1:
import cartan_type
res = cartan_type.CartanType(["A",1,1]).dynkin_diagram()
res._cartan_type = self
return res
from dynkin_diagram import DynkinDiagram_class
g = DynkinDiagram_class(self)
for i in range(1, n):
g.add_edge(i, i+1)
g.set_edge_label(n,n-1,2)
g.add_edge(0,1,2)
return g
def dynkin_diagram(self):
"""
Returns a Dynkin diagram for type E.
EXAMPLES::
sage: e = CartanType(['E',6]).dynkin_diagram()
sage: e
O 2
|
|
O---O---O---O---O
1 3 4 5 6
E6
sage: sorted(e.edges())
[(1, 3, 1), (2, 4, 1), (3, 1, 1), (3, 4, 1), (4, 2, 1), (4, 3, 1), (4, 5, 1), (5, 4, 1), (5, 6, 1), (6, 5, 1)]
sage: e = CartanType(['E',7]).dynkin_diagram()
sage: e
O 2
|
|
O---O---O---O---O---O
1 3 4 5 6 7
E7
sage: sorted(e.edges())
[(1, 3, 1), (2, 4, 1), (3, 1, 1), (3, 4, 1), (4, 2, 1),
(4, 3, 1), (4, 5, 1), (5, 4, 1), (5, 6, 1), (6, 5, 1),
(6, 7, 1), (7, 6, 1)]
sage: e = CartanType(['E',8]).dynkin_diagram()
sage: e
O 2
|
|
O---O---O---O---O---O---O
1 3 4 5 6 7 8
E8
sage: sorted(e.edges())
[(1, 3, 1), (2, 4, 1), (3, 1, 1), (3, 4, 1), (4, 2, 1),
(4, 3, 1), (4, 5, 1), (5, 4, 1), (5, 6, 1), (6, 5, 1),
(6, 7, 1), (7, 6, 1), (7, 8, 1), (8, 7, 1)]
"""
from dynkin_diagram import DynkinDiagram_class
g = DynkinDiagram_class(self)
g.add_edge(1,3)
g.add_edge(2,4)
for i in range(3, self.n):
g.add_edge(i, i+1)
return g
def dynkin_diagram(self):
"""
Returns the extended Dynkin diagram for affine type BC.
EXAMPLES::
sage: c = CartanType(['BC',3,2]).dynkin_diagram()
sage: c
O=<=O---O=<=O
0 1 2 3
BC3~
sage: sorted(c.edges())
[(0, 1, 1), (1, 0, 2), (1, 2, 1), (2, 1, 1), (2, 3, 1), (3, 2, 2)]
sage: c = CartanType(["A", 6, 2]).dynkin_diagram() # should be the same as above; did fail at some point!
sage: c
O=<=O---O=<=O
0 1 2 3
BC3~
sage: sorted(c.edges())
[(0, 1, 1), (1, 0, 2), (1, 2, 1), (2, 1, 1), (2, 3, 1), (3, 2, 2)]
sage: c = CartanType(['BC',2,2]).dynkin_diagram()
sage: c
O=<=O=<=O
0 1 2
BC2~
sage: sorted(c.edges())
[(0, 1, 1), (1, 0, 2), (1, 2, 1), (2, 1, 2)]
sage: c = CartanType(['BC',1,2]).dynkin_diagram()
sage: c
4
O=<=O
0 1
BC1~
sage: sorted(c.edges())
[(0, 1, 1), (1, 0, 4)]
"""
from dynkin_diagram import DynkinDiagram_class
n = self.n
g = DynkinDiagram_class(self)
if n == 1:
g.add_edge(1, 0, 4)
return g
g.add_edge(1, 0, 2)
for i in range(1, n - 1):
g.add_edge(i, i + 1)
g.add_edge(n, n - 1, 2)
return g
def dynkin_diagram(self):
"""
Returns the extended Dynkin diagram for affine type B.
EXAMPLES::
sage: b = CartanType(['B',3,1]).dynkin_diagram()
sage: b
O 0
|
|
O---O=>=O
1 2 3
B3~
sage: sorted(b.edges())
[(0, 2, 1), (1, 2, 1), (2, 0, 1), (2, 1, 1), (2, 3, 2), (3, 2, 1)]
sage: b = CartanType(['B',2,1]).dynkin_diagram(); b
O=>=O=<=O
0 2 1
B2~
sage: sorted(b.edges())
[(0, 2, 2), (1, 2, 2), (2, 0, 1), (2, 1, 1)]
sage: b = CartanType(['B',1,1]).dynkin_diagram(); b
O<=>O
0 1
B1~
sage: sorted(b.edges())
[(0, 1, 2), (1, 0, 2)]
"""
import cartan_type
n = self.n
if n == 1:
res = cartan_type.CartanType(["A", 1, 1]).dynkin_diagram()
res._cartan_type = self
return res
if n == 2:
res = cartan_type.CartanType(["C", 2, 1]).relabel({
0: 0,
1: 2,
2: 1
}).dynkin_diagram()
res._cartan_type = self
return res
from dynkin_diagram import DynkinDiagram_class
g = DynkinDiagram_class(self)
for i in range(1, n):
g.add_edge(i, i + 1)
g.set_edge_label(n - 1, n, 2)
g.add_edge(0, 2)
return g
def dynkin_diagram(self):
"""
Returns a Dynkin diagram for type G.
EXAMPLES::
sage: g = CartanType(['G',2]).dynkin_diagram()
sage: g
3
O=<=O
1 2
G2
sage: sorted(g.edges())
[(1, 2, 1), (2, 1, 3)]
"""
from dynkin_diagram import DynkinDiagram_class
g = DynkinDiagram_class(self)
g.add_edge(1,2)
g.set_edge_label(2,1,3)
return g
def dynkin_diagram(self):
"""
Returns a Dynkin diagram for type F.
EXAMPLES::
sage: f = CartanType(['F',4]).dynkin_diagram()
sage: f
O---O=>=O---O
1 2 3 4
F4
sage: sorted(f.edges())
[(1, 2, 1), (2, 1, 1), (2, 3, 2), (3, 2, 1), (3, 4, 1), (4, 3, 1)]
"""
from dynkin_diagram import DynkinDiagram_class
g = DynkinDiagram_class(self)
for i in range(1, 4):
g.add_edge(i, i+1)
g.set_edge_label(2,3,2)
return g
def dynkin_diagram(self):
"""
Return the Dynkin diagram corresponding to ``self``.
EXAMPLES::
sage: C = CartanMatrix(['A',2])
sage: C.dynkin_diagram()
O---O
1 2
A2
sage: C = CartanMatrix(['F',4,1])
sage: C.dynkin_diagram()
O---O---O=>=O---O
0 1 2 3 4
F4~
sage: C = CartanMatrix([[2,-4],[-4,2]])
sage: C.dynkin_diagram()
Dynkin diagram of rank 2
"""
from sage.combinat.root_system.dynkin_diagram import DynkinDiagram
if self._cartan_type is not None:
return DynkinDiagram(self._cartan_type)
from dynkin_diagram import DynkinDiagram_class
n = self.nrows()
g = DynkinDiagram_class(self)
for i in range(n):
for j in range(n):
if self[i, j] == -1:
g.add_edge(i, j)
elif self[i, j] < -1:
g.add_edge(i, j, -self[i, j])
return g
def dynkin_diagram(self):
"""
Returns a Dynkin diagram for type B.
EXAMPLES::
sage: b = CartanType(['B',3]).dynkin_diagram()
sage: b
O---O=>=O
1 2 3
B3
sage: sorted(b.edges())
[(1, 2, 1), (2, 1, 1), (2, 3, 2), (3, 2, 1)]
sage: b = CartanType(['B',1]).dynkin_diagram()
sage: b
O
1
B1
sage: sorted(b.edges())
[]
"""
from dynkin_diagram import DynkinDiagram_class
n = self.n
g = DynkinDiagram_class(self)
for i in range(1, n):
g.add_edge(i, i + 1)
if n >= 2:
g.set_edge_label(n - 1, n, 2)
return g
def dynkin_diagram(self):
"""
Returns the extended Dynkin diagram for affine type A.
EXAMPLES::
sage: a = CartanType(['A',3,1]).dynkin_diagram()
sage: a
0
O-------+
| |
| |
O---O---O
1 2 3
A3~
sage: sorted(a.edges())
[(0, 1, 1),
(0, 3, 1),
(1, 0, 1),
(1, 2, 1),
(2, 1, 1),
(2, 3, 1),
(3, 0, 1),
(3, 2, 1)]
sage: a = DynkinDiagram(['A',1,1])
sage: a
O<=>O
0 1
A1~
sage: sorted(a.edges())
[(0, 1, 2), (1, 0, 2)]
"""
from dynkin_diagram import DynkinDiagram_class
n = self.n
g = DynkinDiagram_class(self)
if n == 1:
g.add_edge(0, 1, 2)
g.add_edge(1, 0, 2)
else:
for i in range(1, n):
g.add_edge(i, i+1)
g.add_edge(0, 1)
g.add_edge(0, n)
return g
def dynkin_diagram(self):
"""
Returns the extended Dynkin diagram for affine type B.
EXAMPLES::
sage: b = CartanType(['B',3,1]).dynkin_diagram()
sage: b
O 0
|
|
O---O=>=O
1 2 3
B3~
sage: sorted(b.edges())
[(0, 2, 1), (1, 2, 1), (2, 0, 1), (2, 1, 1), (2, 3, 2), (3, 2, 1)]
sage: b = CartanType(['B',2,1]).dynkin_diagram(); b
O=>=O=<=O
0 2 1
B2~
sage: sorted(b.edges())
[(0, 2, 2), (1, 2, 2), (2, 0, 1), (2, 1, 1)]
sage: b = CartanType(['B',1,1]).dynkin_diagram(); b
O<=>O
0 1
B1~
sage: sorted(b.edges())
[(0, 1, 2), (1, 0, 2)]
"""
import cartan_type
n = self.n
if n == 1:
res = cartan_type.CartanType(["A",1,1]).dynkin_diagram()
res._cartan_type = self
return res
if n == 2:
res = cartan_type.CartanType(["C",2,1]).relabel({0:0, 1:2, 2:1}).dynkin_diagram()
res._cartan_type = self
return res
from dynkin_diagram import DynkinDiagram_class
g = DynkinDiagram_class(self)
for i in range(1, n):
g.add_edge(i, i+1)
g.set_edge_label(n-1, n, 2)
g.add_edge(0,2)
return g
def dynkin_diagram(self):
"""
Returns a Dynkin diagram for type D.
EXAMPLES::
sage: d = CartanType(['D',5]).dynkin_diagram(); d
O 5
|
|
O---O---O---O
1 2 3 4
D5
sage: sorted(d.edges())
[(1, 2, 1), (2, 1, 1), (2, 3, 1), (3, 2, 1), (3, 4, 1), (3, 5, 1), (4, 3, 1), (5, 3, 1)]
sage: d = CartanType(['D',4]).dynkin_diagram(); d
O 4
|
|
O---O---O
1 2 3
D4
sage: sorted(d.edges())
[(1, 2, 1), (2, 1, 1), (2, 3, 1), (2, 4, 1), (3, 2, 1), (4, 2, 1)]
sage: d = CartanType(['D',3]).dynkin_diagram(); d
O 3
|
|
O---O
1 2
D3
sage: sorted(d.edges())
[(1, 2, 1), (1, 3, 1), (2, 1, 1), (3, 1, 1)]
sage: d = CartanType(['D',2]).dynkin_diagram(); d
O O
1 2
D2
sage: sorted(d.edges())
[]
"""
from dynkin_diagram import DynkinDiagram_class
g = DynkinDiagram_class(self)
n = self.n
if n >= 3:
for i in range(1, n - 1):
g.add_edge(i, i + 1)
g.add_edge(n - 2, n)
return g
def dynkin_diagram(self):
"""
Returns the extended Dynkin diagram for affine type A.
EXAMPLES::
sage: a = CartanType(['A',3,1]).dynkin_diagram()
sage: a
0
O-------+
| |
| |
O---O---O
1 2 3
A3~
sage: sorted(a.edges())
[(0, 1, 1),
(0, 3, 1),
(1, 0, 1),
(1, 2, 1),
(2, 1, 1),
(2, 3, 1),
(3, 0, 1),
(3, 2, 1)]
sage: a = DynkinDiagram(['A',1,1])
sage: a
O<=>O
0 1
A1~
sage: sorted(a.edges())
[(0, 1, 2), (1, 0, 2)]
"""
from dynkin_diagram import DynkinDiagram_class
n = self.n
g = DynkinDiagram_class(self)
if n == 1:
g.add_edge(0, 1, 2)
g.add_edge(1, 0, 2)
else:
for i in range(1, n):
g.add_edge(i, i+1)
g.add_edge(0, 1)
g.add_edge(0, n)
return g
def dynkin_diagram(self):
"""
Returns the extended Dynkin diagram for type G.
EXAMPLES::
sage: g = CartanType(['G',2,1]).dynkin_diagram()
sage: g
3
O=<=O---O
1 2 0
G2~
sage: sorted(g.edges())
[(0, 2, 1), (1, 2, 1), (2, 0, 1), (2, 1, 3)]
"""
from dynkin_diagram import DynkinDiagram_class
g = DynkinDiagram_class(self)
g.add_edge(1, 2)
g.set_edge_label(2,1,3)
g.add_edge(0, 2)
return g
def dynkin_diagram(self):
"""
Returns a Dynkin diagram for type E.
EXAMPLES::
sage: e = CartanType(['E',6]).dynkin_diagram()
sage: e
O 2
|
|
O---O---O---O---O
1 3 4 5 6
E6
sage: sorted(e.edges())
[(1, 3, 1), (2, 4, 1), (3, 1, 1), (3, 4, 1), (4, 2, 1), (4, 3, 1), (4, 5, 1), (5, 4, 1), (5, 6, 1), (6, 5, 1)]
sage: e = CartanType(['E',7]).dynkin_diagram()
sage: e
O 2
|
|
O---O---O---O---O---O
1 3 4 5 6 7
E7
sage: sorted(e.edges())
[(1, 3, 1), (2, 4, 1), (3, 1, 1), (3, 4, 1), (4, 2, 1),
(4, 3, 1), (4, 5, 1), (5, 4, 1), (5, 6, 1), (6, 5, 1),
(6, 7, 1), (7, 6, 1)]
sage: e = CartanType(['E',8]).dynkin_diagram()
sage: e
O 2
|
|
O---O---O---O---O---O---O
1 3 4 5 6 7 8
E8
sage: sorted(e.edges())
[(1, 3, 1), (2, 4, 1), (3, 1, 1), (3, 4, 1), (4, 2, 1),
(4, 3, 1), (4, 5, 1), (5, 4, 1), (5, 6, 1), (6, 5, 1),
(6, 7, 1), (7, 6, 1), (7, 8, 1), (8, 7, 1)]
"""
from dynkin_diagram import DynkinDiagram_class
g = DynkinDiagram_class(self)
g.add_edge(1,3)
g.add_edge(2,4)
for i in range(3, self.n):
g.add_edge(i, i+1)
return g
def dynkin_diagram(self):
"""
Returns the extended Dynkin diagram for affine type F.
EXAMPLES::
sage: f = CartanType(['F', 4, 1]).dynkin_diagram()
sage: f
O---O---O=>=O---O
0 1 2 3 4
F4~
sage: sorted(f.edges())
[(0, 1, 1), (1, 0, 1), (1, 2, 1), (2, 1, 1), (2, 3, 2), (3, 2, 1), (3, 4, 1), (4, 3, 1)]
"""
from dynkin_diagram import DynkinDiagram_class
g = DynkinDiagram_class(self)
for i in range(1, 4):
g.add_edge(i, i+1)
g.set_edge_label(2,3,2)
g.add_edge(0, 1)
return g
def dynkin_diagram(self):
"""
Returns the extended Dynkin diagram for affine type F.
EXAMPLES::
sage: f = DynkinDiagram(['F', 4, 1])
sage: f
O---O---O=>=O---O
0 1 2 3 4
F4~
sage: sorted(f.edges())
[(0, 1, 1), (1, 0, 1), (1, 2, 1), (2, 1, 1), (2, 3, 2), (3, 2, 1), (3, 4, 1), (4, 3, 1)]
"""
from dynkin_diagram import DynkinDiagram_class
g = DynkinDiagram_class(self)
for i in range(1, 4):
g.add_edge(i, i+1)
g.set_edge_label(2,3,2)
g.add_edge(0, 1)
return g
def dynkin_diagram(self):
"""
Returns a Dynkin diagram for type G.
EXAMPLES::
sage: g = CartanType(['G',2]).dynkin_diagram()
sage: g
3
O=<=O
1 2
G2
sage: sorted(g.edges())
[(1, 2, 1), (2, 1, 3)]
"""
from dynkin_diagram import DynkinDiagram_class
g = DynkinDiagram_class(self)
g.add_edge(1, 2)
g.set_edge_label(2, 1, 3)
return g
def dynkin_diagram(self):
"""
Returns a Dynkin diagram for type F.
EXAMPLES::
sage: f = DynkinDiagram(['F',4])
sage: f
O---O=>=O---O
1 2 3 4
F4
sage: sorted(f.edges())
[(1, 2, 1), (2, 1, 1), (2, 3, 2), (3, 2, 1), (3, 4, 1), (4, 3, 1)]
"""
from dynkin_diagram import DynkinDiagram_class
g = DynkinDiagram_class(self)
for i in range(1, 4):
g.add_edge(i, i + 1)
g.set_edge_label(2, 3, 2)
return g
def dynkin_diagram(self):
"""
Returns the extended Dynkin diagram for affine type E.
EXAMPLES::
sage: e = CartanType(['E', 6, 1]).dynkin_diagram()
sage: e
O 0
|
|
O 2
|
|
O---O---O---O---O
1 3 4 5 6
E6~
sage: sorted(e.edges())
[(0, 2, 1),
(1, 3, 1),
(2, 0, 1),
(2, 4, 1),
(3, 1, 1),
(3, 4, 1),
(4, 2, 1),
(4, 3, 1),
(4, 5, 1),
(5, 4, 1),
(5, 6, 1),
(6, 5, 1)]
sage: e = CartanType(['E', 7, 1]).dynkin_diagram()
sage: e
O 2
|
|
O---O---O---O---O---O---O
0 1 3 4 5 6 7
E7~
sage: sorted(e.edges())
[(0, 1, 1), (1, 0, 1), (1, 3, 1), (2, 4, 1), (3, 1, 1), (3, 4, 1),
(4, 2, 1), (4, 3, 1), (4, 5, 1), (5, 4, 1), (5, 6, 1),
(6, 5, 1), (6, 7, 1), (7, 6, 1)]
sage: e = CartanType(['E', 8, 1]).dynkin_diagram()
sage: e
O 2
|
|
O---O---O---O---O---O---O---O
1 3 4 5 6 7 8 0
E8~
sage: sorted(e.edges())
[(0, 8, 1), (1, 3, 1), (2, 4, 1), (3, 1, 1), (3, 4, 1),
(4, 2, 1), (4, 3, 1), (4, 5, 1), (5, 4, 1), (5, 6, 1),
(6, 5, 1), (6, 7, 1), (7, 6, 1), (7, 8, 1), (8, 0, 1), (8, 7, 1)]
"""
from dynkin_diagram import DynkinDiagram_class
n = self.n
g = DynkinDiagram_class(self)
g.add_edge(1, 3)
g.add_edge(2, 4)
for i in range(3, n):
g.add_edge(i, i + 1)
if n == 6:
g.add_edge(0, 2)
elif n == 7:
g.add_edge(0, 1)
elif n == 8:
g.add_edge(0, 8)
else:
raise ValueError("Invalid Cartan Type for Type E affine")
return g
def dynkin_diagram(self):
"""
Returns the extended Dynkin diagram for affine type E.
EXAMPLES::
sage: e = CartanType(['E', 6, 1]).dynkin_diagram()
sage: e
O 0
|
|
O 2
|
|
O---O---O---O---O
1 3 4 5 6
E6~
sage: sorted(e.edges())
[(0, 2, 1),
(1, 3, 1),
(2, 0, 1),
(2, 4, 1),
(3, 1, 1),
(3, 4, 1),
(4, 2, 1),
(4, 3, 1),
(4, 5, 1),
(5, 4, 1),
(5, 6, 1),
(6, 5, 1)]
sage: e = CartanType(['E', 7, 1]).dynkin_diagram()
sage: e
O 2
|
|
O---O---O---O---O---O---O
0 1 3 4 5 6 7
E7~
sage: sorted(e.edges())
[(0, 1, 1), (1, 0, 1), (1, 3, 1), (2, 4, 1), (3, 1, 1), (3, 4, 1),
(4, 2, 1), (4, 3, 1), (4, 5, 1), (5, 4, 1), (5, 6, 1),
(6, 5, 1), (6, 7, 1), (7, 6, 1)]
sage: e = CartanType(['E', 8, 1]).dynkin_diagram()
sage: e
O 2
|
|
O---O---O---O---O---O---O---O
1 3 4 5 6 7 8 0
E8~
sage: sorted(e.edges())
[(0, 8, 1), (1, 3, 1), (2, 4, 1), (3, 1, 1), (3, 4, 1),
(4, 2, 1), (4, 3, 1), (4, 5, 1), (5, 4, 1), (5, 6, 1),
(6, 5, 1), (6, 7, 1), (7, 6, 1), (7, 8, 1), (8, 0, 1), (8, 7, 1)]
"""
from dynkin_diagram import DynkinDiagram_class
n = self.n
g = DynkinDiagram_class(self)
g.add_edge(1, 3)
g.add_edge(2, 4)
for i in range(3, n):
g.add_edge(i, i + 1)
if n == 6:
g.add_edge(0, 2)
elif n == 7:
g.add_edge(0, 1)
elif n == 8:
g.add_edge(0, 8)
else:
assert False, "Invalid Cartan Type for Type E affine"
return g
def dynkin_diagram(self):
"""
Returns the extended Dynkin diagram for affine type D.
EXAMPLES::
sage: d = CartanType(['D', 6, 1]).dynkin_diagram()
sage: d
0 O O 6
| |
| |
O---O---O---O---O
1 2 3 4 5
D6~
sage: sorted(d.edges())
[(0, 2, 1), (1, 2, 1), (2, 0, 1), (2, 1, 1), (2, 3, 1),
(3, 2, 1), (3, 4, 1), (4, 3, 1), (4, 5, 1), (4, 6, 1), (5, 4, 1), (6, 4, 1)]
sage: d = CartanType(['D', 4, 1]).dynkin_diagram()
sage: d
O 4
|
|
O---O---O
1 |2 3
|
O 0
D4~
sage: sorted(d.edges())
[(0, 2, 1),
(1, 2, 1),
(2, 0, 1),
(2, 1, 1),
(2, 3, 1),
(2, 4, 1),
(3, 2, 1),
(4, 2, 1)]
sage: d = CartanType(['D', 3, 1]).dynkin_diagram()
sage: d
0
O-------+
| |
| |
O---O---O
3 1 2
D3~
sage: sorted(d.edges())
[(0, 2, 1), (0, 3, 1), (1, 2, 1), (1, 3, 1), (2, 0, 1), (2, 1, 1), (3, 0, 1), (3, 1, 1)]
"""
from dynkin_diagram import DynkinDiagram_class
n = self.n
if n == 3:
import cartan_type
res = cartan_type.CartanType(["A", 3, 1]).relabel({
0: 0,
1: 3,
2: 1,
3: 2
}).dynkin_diagram()
res._cartan_type = self
return res
g = DynkinDiagram_class(self)
for i in range(1, n - 1):
g.add_edge(i, i + 1)
g.add_edge(n - 2, n)
g.add_edge(0, 2)
return g
def dynkin_diagram(self):
"""
Returns the extended Dynkin diagram for affine type D.
EXAMPLES::
sage: d = CartanType(['D', 6, 1]).dynkin_diagram()
sage: d
0 O O 6
| |
| |
O---O---O---O---O
1 2 3 4 5
D6~
sage: sorted(d.edges())
[(0, 2, 1), (1, 2, 1), (2, 0, 1), (2, 1, 1), (2, 3, 1),
(3, 2, 1), (3, 4, 1), (4, 3, 1), (4, 5, 1), (4, 6, 1), (5, 4, 1), (6, 4, 1)]
sage: d = CartanType(['D', 4, 1]).dynkin_diagram()
sage: d
O 4
|
|
O---O---O
1 |2 3
|
O 0
D4~
sage: sorted(d.edges())
[(0, 2, 1),
(1, 2, 1),
(2, 0, 1),
(2, 1, 1),
(2, 3, 1),
(2, 4, 1),
(3, 2, 1),
(4, 2, 1)]
sage: d = CartanType(['D', 3, 1]).dynkin_diagram()
sage: d
0
O-------+
| |
| |
O---O---O
3 1 2
D3~
sage: sorted(d.edges())
[(0, 2, 1), (0, 3, 1), (1, 2, 1), (1, 3, 1), (2, 0, 1), (2, 1, 1), (3, 0, 1), (3, 1, 1)]
"""
from dynkin_diagram import DynkinDiagram_class
n = self.n
if n == 3:
import cartan_type
res = cartan_type.CartanType(["A",3,1]).relabel({0:0, 1:3, 2:1, 3: 2}).dynkin_diagram()
res._cartan_type = self
return res
g = DynkinDiagram_class(self)
for i in range(1, n-1):
g.add_edge(i, i+1)
g.add_edge(n-2,n)
g.add_edge(0,2)
return g
