def _latex_dynkin_diagram(self, label=lambda x: x, node_dist=2, dual=False):
r"""
Return a latex representation of the Dynkin diagram.
EXAMPLES::
sage: print CartanType(['B',4,1])._latex_dynkin_diagram()
\draw (0,0.7 cm) -- (2 cm,0);
\draw (0,-0.7 cm) -- (2 cm,0);
\draw (2 cm,0) -- (4 cm,0);
\draw (4 cm, 0.1 cm) -- +(2 cm,0);
\draw (4 cm, -0.1 cm) -- +(2 cm,0);
\draw[shift={(5.2, 0)}, rotate=0] (135 : 0.45cm) -- (0,0) -- (-135 : 0.45cm);
\draw[fill=white] (0, 0.7 cm) circle (.25cm) node[left=3pt]{$0$};
\draw[fill=white] (0, -0.7 cm) circle (.25cm) node[left=3pt]{$1$};
\draw[fill=white] (2 cm, 0) circle (.25cm) node[below=4pt]{$2$};
\draw[fill=white] (4 cm, 0) circle (.25cm) node[below=4pt]{$3$};
\draw[fill=white] (6 cm, 0) circle (.25cm) node[below=4pt]{$4$};
sage: print CartanType(['B',4,1]).dual()._latex_dynkin_diagram()
\draw (0,0.7 cm) -- (2 cm,0);
\draw (0,-0.7 cm) -- (2 cm,0);
\draw (2 cm,0) -- (4 cm,0);
\draw (4 cm, 0.1 cm) -- +(2 cm,0);
\draw (4 cm, -0.1 cm) -- +(2 cm,0);
\draw[shift={(4.8, 0)}, rotate=180] (135 : 0.45cm) -- (0,0) -- (-135 : 0.45cm);
\draw[fill=white] (0, 0.7 cm) circle (.25cm) node[left=3pt]{$0$};
\draw[fill=white] (0, -0.7 cm) circle (.25cm) node[left=3pt]{$1$};
\draw[fill=white] (2 cm, 0) circle (.25cm) node[below=4pt]{$2$};
\draw[fill=white] (4 cm, 0) circle (.25cm) node[below=4pt]{$3$};
\draw[fill=white] (6 cm, 0) circle (.25cm) node[below=4pt]{$4$};
"""
if self.n == 1:
import cartan_type
return cartan_type.CartanType(["A",1,1])._latex_dynkin_diagram(label, node_dist)
elif self.n == 2:
import cartan_type
return cartan_type.CartanType(["C",2,1])._latex_dynkin_diagram(label, node_dist, dual)
if self.global_options('mark_special_node') in ['latex', 'both']:
special_fill = 'black'
else:
special_fill = 'white'
n = self.n
single_end = (n-2)*node_dist # Where the single line ends
ret = "\\draw (0,0.7 cm) -- (%s cm,0);\n"%node_dist
ret += "\\draw (0,-0.7 cm) -- (%s cm,0);\n"%node_dist
ret += "\\draw (%s cm,0) -- (%s cm,0);\n"%(node_dist, single_end)
ret += "\\draw (%s cm, 0.1 cm) -- +(%s cm,0);\n"%(single_end, node_dist)
ret += "\\draw (%s cm, -0.1 cm) -- +(%s cm,0);\n"%(single_end, node_dist)
if dual:
ret += self._latex_draw_arrow_tip(single_end+0.5*node_dist-0.2, 0, 180)
else:
ret += self._latex_draw_arrow_tip(single_end+0.5*node_dist+0.2, 0, 0)
ret += "\\draw[fill=%s] (0, 0.7 cm) circle (.25cm) node[left=3pt]{$%s$};\n"%(special_fill, label(0))
ret += "\\draw[fill=white] (0, -0.7 cm) circle (.25cm) node[left=3pt]{$%s$};\n"%label(1)
for i in range(1, n-1):
ret += "\\draw[fill=white] (%s cm, 0) circle (.25cm) node[below=4pt]{$%s$};\n"%(i*node_dist, label(i+1))
ret += "\\draw[fill=white] (%s cm, 0) circle (.25cm) node[below=4pt]{$%s$};"%((n-1)*node_dist, label(n))
return ret
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 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 ascii_art(self, label=lambda i: i, node=None):
"""
Return an ascii art representation of the extended Dynkin diagram.
TESTS::
sage: print CartanType(['D',6,1]).ascii_art(label = lambda x: x+2)
2 O O 8
| |
| |
O---O---O---O---O
3 4 5 6 7
sage: print CartanType(['D',4,1]).ascii_art(label = lambda x: x+2)
O 6
|
|
O---O---O
3 |4 5
|
O 2
sage: print CartanType(['D',3,1]).ascii_art(label = lambda x: x+2)
2
O-------+
| |
| |
O---O---O
5 3 4
"""
if node is None:
node = self._ascii_art_node
n = self.n
if n == 3:
import cartan_type
return cartan_type.CartanType(["A", 3, 1]).relabel({
0: 0,
1: 3,
2: 1,
3: 2
}).ascii_art(label, node)
if n == 4:
ret = " {} {}\n".format(node(label(4)),
label(4)) + " |\n |\n"
ret += "{}---{}---{}\n".format(node(label(1)), node(label(2)),
node(label(3)))
ret += "{!s:4}|{!s:3}{!s:4}\n".format(label(1), label(2), label(3))
ret += " |\n {} {}".format(node(label(0)), label(0))
return ret
ret = "{!s:>3} {}".format(label(0), node(label(0)))
ret += (4 *
(n - 4) - 1) * " " + "{} {}\n".format(node(label(n)), label(n))
ret += " |" + (4 * (n - 4) - 1) * " " + "|\n"
ret += " |" + (4 * (n - 4) - 1) * " " + "|\n"
ret += "---".join(node(label(i)) for i in range(1, n))
ret += '\n' + "".join("{!s:4}".format(label(i)) for i in range(1, n))
return ret
def classical(self):
"""
Returns the classical Cartan type associated with self
sage: CartanType(["BC", 3, 2]).classical()
['C', 3]
"""
import cartan_type
return cartan_type.CartanType(["C", self.n])
def _latex_dynkin_diagram(self, label=lambda x: x, node_dist=2, dual=False):
r"""
Return a latex representation of the Dynkin diagram.
EXAMPLES::
sage: print CartanType(['C',4,1])._latex_dynkin_diagram()
\draw (0, 0.1 cm) -- +(2 cm,0);
\draw (0, -0.1 cm) -- +(2 cm,0);
\draw[shift={(1.2, 0)}, rotate=0] (135 : 0.45cm) -- (0,0) -- (-135 : 0.45cm);
{
\pgftransformxshift{2 cm}
\draw (0 cm,0) -- (4 cm,0);
\draw (4 cm, 0.1 cm) -- +(2 cm,0);
\draw (4 cm, -0.1 cm) -- +(2 cm,0);
\draw[shift={(4.8, 0)}, rotate=180] (135 : 0.45cm) -- (0,0) -- (-135 : 0.45cm);
\draw[fill=white] (0 cm, 0) circle (.25cm) node[below=4pt]{$1$};
\draw[fill=white] (2 cm, 0) circle (.25cm) node[below=4pt]{$2$};
\draw[fill=white] (4 cm, 0) circle (.25cm) node[below=4pt]{$3$};
\draw[fill=white] (6 cm, 0) circle (.25cm) node[below=4pt]{$4$};
}
\draw[fill=white] (0, 0) circle (.25cm) node[below=4pt]{$0$};
sage: print CartanType(['C',4,1]).dual()._latex_dynkin_diagram()
\draw (0, 0.1 cm) -- +(2 cm,0);
\draw (0, -0.1 cm) -- +(2 cm,0);
\draw[shift={(0.8, 0)}, rotate=180] (135 : 0.45cm) -- (0,0) -- (-135 : 0.45cm);
{
\pgftransformxshift{2 cm}
\draw (0 cm,0) -- (4 cm,0);
\draw (4 cm, 0.1 cm) -- +(2 cm,0);
\draw (4 cm, -0.1 cm) -- +(2 cm,0);
\draw[shift={(5.2, 0)}, rotate=0] (135 : 0.45cm) -- (0,0) -- (-135 : 0.45cm);
\draw[fill=white] (0 cm, 0) circle (.25cm) node[below=4pt]{$1$};
\draw[fill=white] (2 cm, 0) circle (.25cm) node[below=4pt]{$2$};
\draw[fill=white] (4 cm, 0) circle (.25cm) node[below=4pt]{$3$};
\draw[fill=white] (6 cm, 0) circle (.25cm) node[below=4pt]{$4$};
}
\draw[fill=white] (0, 0) circle (.25cm) node[below=4pt]{$0$};
"""
if self.n == 1:
import cartan_type
return cartan_type.CartanType(["A",1,1])._latex_dynkin_diagram(label, node_dist)
if self.global_options('mark_special_node') in ['latex', 'both']:
special_fill = 'black'
else:
special_fill = 'white'
ret = "\\draw (0, 0.1 cm) -- +(%s cm,0);\n"%node_dist
ret += "\\draw (0, -0.1 cm) -- +(%s cm,0);\n"%node_dist
if dual:
ret += self._latex_draw_arrow_tip(0.5*node_dist-0.2, 0, 180)
else:
ret += self._latex_draw_arrow_tip(0.5*node_dist+0.2, 0, 0)
ret += "{\n\\pgftransformxshift{%s cm}\n"%node_dist
ret += self.classical()._latex_dynkin_diagram(label, node_dist, dual)
ret += "\n}\n\\draw[fill=%s] (0, 0) circle (.25cm) node[below=4pt]{$%s$};"%(special_fill, label(0))
return ret
def dual(self):
"""
Types B and C are in duality:
EXAMPLES::
sage: CartanType(["C", 3]).dual()
['B', 3]
"""
import cartan_type
return cartan_type.CartanType(["B", self.n])
def ascii_art(self, label=lambda x: x):
"""
Returns a ascii art representation of the extended Dynkin diagram
TESTS::
sage: print CartanType(['D',6,1]).ascii_art(label = lambda x: x+2)
2 O O 8
| |
| |
O---O---O---O---O
3 4 5 6 7
sage: print CartanType(['D',4,1]).ascii_art(label = lambda x: x+2)
O 6
|
|
O---O---O
3 |4 5
|
O 2
sage: print CartanType(['D',3,1]).ascii_art(label = lambda x: x+2)
2
O-------+
| |
| |
O---O---O
5 3 4
"""
n = self.n
if n == 3:
import cartan_type
return cartan_type.CartanType(["A", 3, 1]).relabel({
0: 0,
1: 3,
2: 1,
3: 2
}).ascii_art(label)
if self.global_options('mark_special_node') in ['printing', 'both']:
special_str = self.global_options('special_node_str')
else:
special_str = 'O'
if n == 4:
return " " + special_str + " %s\n |\n |\nO---O---O\n%s |%s %s\n |\n O %s" % tuple(
label(i) for i in (4, 1, 2, 3, 0))
ret = " %s O" % label(0) + (4 * (n - 4) -
1) * " " + "O %s" % label(n) + "\n"
ret += " |" + (4 * (n - 4) - 1) * " " + "|\n"
ret += " |" + (4 * (n - 4) - 1) * " " + "|\n"
ret += (n - 2) * "O---" + "O\n"
ret += " ".join("%s" % label(i) for i in range(1, n))
return ret
def _latex_dynkin_diagram(self,
label=lambda x: x,
node_dist=2,
dual=False):
r"""
Return a latex representation of the Dynkin diagram.
EXAMPLES::
sage: print CartanType(['D',4,1])._latex_dynkin_diagram()
\draw (0,0.7 cm) -- (2 cm,0);
\draw (0,-0.7 cm) -- (2 cm,0);
\draw (2 cm,0) -- (2 cm,0);
\draw (2 cm,0) -- (4 cm,0.7 cm);
\draw (2 cm,0) -- (4 cm,-0.7 cm);
\draw[fill=white] (0, 0.7 cm) circle (.25cm) node[left=3pt]{$0$};
\draw[fill=white] (0, -0.7 cm) circle (.25cm) node[left=3pt]{$1$};
\draw[fill=white] (2 cm, 0) circle (.25cm) node[below=4pt]{$2$};
\draw[fill=white] (4 cm, 0.7 cm) circle (.25cm) node[right=3pt]{$4$};
\draw[fill=white] (4 cm, -0.7 cm) circle (.25cm) node[right=3pt]{$3$};
"""
n = self.n
if n == 3:
import cartan_type
relabel = {0: label(0), 1: label(3), 2: label(1), 3: label(2)}
return cartan_type.CartanType([
"A", 3, 1
]).relabel(relabel)._latex_dynkin_diagram(node_dist=node_dist)
if self.global_options('mark_special_node') in ['latex', 'both']:
special_fill = 'black'
else:
special_fill = 'white'
rt_most = (n - 2) * node_dist
center_point = rt_most - node_dist
ret = "\\draw (0,0.7 cm) -- (%s cm,0);\n" % node_dist
ret += "\\draw (0,-0.7 cm) -- (%s cm,0);\n" % node_dist
ret += "\\draw (%s cm,0) -- (%s cm,0);\n" % (node_dist, center_point)
ret += "\\draw (%s cm,0) -- (%s cm,0.7 cm);\n" % (center_point,
rt_most)
ret += "\\draw (%s cm,0) -- (%s cm,-0.7 cm);\n" % (center_point,
rt_most)
ret += "\\draw[fill=%s] (0, 0.7 cm) circle (.25cm) node[left=3pt]{$%s$};\n" % (
special_fill, label(0))
ret += "\\draw[fill=white] (0, -0.7 cm) circle (.25cm) node[left=3pt]{$%s$};\n" % label(
1)
for i in range(1, self.n - 2):
ret += "\\draw[fill=white] (%s cm, 0) circle (.25cm) node[below=4pt]{$%s$};\n" % (
i * node_dist, label(i + 1))
ret += "\\draw[fill=white] (%s cm, 0.7 cm) circle (.25cm) node[right=3pt]{$%s$};\n" % (
rt_most, label(n))
ret += "\\draw[fill=white] (%s cm, -0.7 cm) circle (.25cm) node[right=3pt]{$%s$};" % (
rt_most, label(n - 1))
return ret
def basic_untwisted(self):
r"""
Return the basic untwisted Cartan type associated with this affine
Cartan type.
Given an affine type `X_n^{(r)}`, the basic untwisted type is `X_n`.
In other words, it is the classical Cartan type that is twisted to
obtain ``self``.
EXAMPLES::
sage: CartanType(['A', 2, 2]).basic_untwisted()
['A', 2]
sage: CartanType(['A', 4, 2]).basic_untwisted()
['A', 4]
sage: CartanType(['BC', 4, 2]).basic_untwisted()
['A', 8]
"""
import cartan_type
return cartan_type.CartanType(["A", 2 * self.n])
def cartan_matrix(t):
"""
Returns the Cartan matrix corresponding to type t.
EXAMPLES::
sage: cartan_matrix(['A', 4])
[ 2 -1 0 0]
[-1 2 -1 0]
[ 0 -1 2 -1]
[ 0 0 -1 2]
sage: cartan_matrix(['B', 6])
[ 2 -1 0 0 0 0]
[-1 2 -1 0 0 0]
[ 0 -1 2 -1 0 0]
[ 0 0 -1 2 -1 0]
[ 0 0 0 -1 2 -1]
[ 0 0 0 0 -2 2]
sage: cartan_matrix(['C', 4])
[ 2 -1 0 0]
[-1 2 -1 0]
[ 0 -1 2 -2]
[ 0 0 -1 2]
sage: cartan_matrix(['D', 6])
[ 2 -1 0 0 0 0]
[-1 2 -1 0 0 0]
[ 0 -1 2 -1 0 0]
[ 0 0 -1 2 -1 -1]
[ 0 0 0 -1 2 0]
[ 0 0 0 -1 0 2]
sage: cartan_matrix(['E',6])
[ 2 0 -1 0 0 0]
[ 0 2 0 -1 0 0]
[-1 0 2 -1 0 0]
[ 0 -1 -1 2 -1 0]
[ 0 0 0 -1 2 -1]
[ 0 0 0 0 -1 2]
sage: cartan_matrix(['E',7])
[ 2 0 -1 0 0 0 0]
[ 0 2 0 -1 0 0 0]
[-1 0 2 -1 0 0 0]
[ 0 -1 -1 2 -1 0 0]
[ 0 0 0 -1 2 -1 0]
[ 0 0 0 0 -1 2 -1]
[ 0 0 0 0 0 -1 2]
sage: cartan_matrix(['E', 8])
[ 2 0 -1 0 0 0 0 0]
[ 0 2 0 -1 0 0 0 0]
[-1 0 2 -1 0 0 0 0]
[ 0 -1 -1 2 -1 0 0 0]
[ 0 0 0 -1 2 -1 0 0]
[ 0 0 0 0 -1 2 -1 0]
[ 0 0 0 0 0 -1 2 -1]
[ 0 0 0 0 0 0 -1 2]
sage: cartan_matrix(['F', 4])
[ 2 -1 0 0]
[-1 2 -1 0]
[ 0 -2 2 -1]
[ 0 0 -1 2]
This is different from MuPAD-Combinat, due to different node
convention?
::
sage: cartan_matrix(['G', 2])
[ 2 -3]
[-1 2]
sage: cartan_matrix(['A',1,1])
[ 2 -2]
[-2 2]
sage: cartan_matrix(['A', 3, 1])
[ 2 -1 0 -1]
[-1 2 -1 0]
[ 0 -1 2 -1]
[-1 0 -1 2]
sage: cartan_matrix(['B', 3, 1])
[ 2 0 -1 0]
[ 0 2 -1 0]
[-1 -1 2 -1]
[ 0 0 -2 2]
sage: cartan_matrix(['C', 3, 1])
[ 2 -1 0 0]
[-2 2 -1 0]
[ 0 -1 2 -2]
[ 0 0 -1 2]
sage: cartan_matrix(['D', 4, 1])
[ 2 0 -1 0 0]
[ 0 2 -1 0 0]
[-1 -1 2 -1 -1]
[ 0 0 -1 2 0]
[ 0 0 -1 0 2]
sage: cartan_matrix(['E', 6, 1])
[ 2 0 -1 0 0 0 0]
[ 0 2 0 -1 0 0 0]
[-1 0 2 0 -1 0 0]
[ 0 -1 0 2 -1 0 0]
[ 0 0 -1 -1 2 -1 0]
[ 0 0 0 0 -1 2 -1]
[ 0 0 0 0 0 -1 2]
sage: cartan_matrix(['E', 7, 1])
[ 2 -1 0 0 0 0 0 0]
[-1 2 0 -1 0 0 0 0]
[ 0 0 2 0 -1 0 0 0]
[ 0 -1 0 2 -1 0 0 0]
[ 0 0 -1 -1 2 -1 0 0]
[ 0 0 0 0 -1 2 -1 0]
[ 0 0 0 0 0 -1 2 -1]
[ 0 0 0 0 0 0 -1 2]
sage: cartan_matrix(['E', 8, 1])
[ 2 0 0 0 0 0 0 0 -1]
[ 0 2 0 -1 0 0 0 0 0]
[ 0 0 2 0 -1 0 0 0 0]
[ 0 -1 0 2 -1 0 0 0 0]
[ 0 0 -1 -1 2 -1 0 0 0]
[ 0 0 0 0 -1 2 -1 0 0]
[ 0 0 0 0 0 -1 2 -1 0]
[ 0 0 0 0 0 0 -1 2 -1]
[-1 0 0 0 0 0 0 -1 2]
sage: cartan_matrix(['F', 4, 1])
[ 2 -1 0 0 0]
[-1 2 -1 0 0]
[ 0 -1 2 -1 0]
[ 0 0 -2 2 -1]
[ 0 0 0 -1 2]
sage: cartan_matrix(['G', 2, 1])
[ 2 0 -1]
[ 0 2 -3]
[-1 -1 2]
"""
t = cartan_type.CartanType(t)
dynkin_diagram = t.dynkin_diagram()
index_set = t.index_set()
MS = MatrixSpace(ZZ, len(index_set), sparse=True)
m = MS(0)
for i in range(len(index_set)):
for j in range(len(index_set)):
m[i, j] = dynkin_diagram[index_set[i], index_set[j]]
return m
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 _latex_dynkin_diagram(self,
label=lambda i: i,
node=None,
node_dist=2,
dual=False):
r"""
Return a latex representation of the Dynkin diagram.
EXAMPLES::
sage: print CartanType(['C',4,1])._latex_dynkin_diagram()
\draw (0, 0.1 cm) -- +(2 cm,0);
\draw (0, -0.1 cm) -- +(2 cm,0);
\draw[shift={(1.2, 0)}, rotate=0] (135 : 0.45cm) -- (0,0) -- (-135 : 0.45cm);
{
\pgftransformxshift{2 cm}
\draw (0 cm,0) -- (4 cm,0);
\draw (4 cm, 0.1 cm) -- +(2 cm,0);
\draw (4 cm, -0.1 cm) -- +(2 cm,0);
\draw[shift={(4.8, 0)}, rotate=180] (135 : 0.45cm) -- (0,0) -- (-135 : 0.45cm);
\draw[fill=white] (0 cm, 0 cm) circle (.25cm) node[below=4pt]{$1$};
\draw[fill=white] (2 cm, 0 cm) circle (.25cm) node[below=4pt]{$2$};
\draw[fill=white] (4 cm, 0 cm) circle (.25cm) node[below=4pt]{$3$};
\draw[fill=white] (6 cm, 0 cm) circle (.25cm) node[below=4pt]{$4$};
}
\draw[fill=white] (0 cm, 0 cm) circle (.25cm) node[below=4pt]{$0$};
sage: print CartanType(['C',4,1]).dual()._latex_dynkin_diagram()
\draw (0, 0.1 cm) -- +(2 cm,0);
\draw (0, -0.1 cm) -- +(2 cm,0);
\draw[shift={(0.8, 0)}, rotate=180] (135 : 0.45cm) -- (0,0) -- (-135 : 0.45cm);
{
\pgftransformxshift{2 cm}
\draw (0 cm,0) -- (4 cm,0);
\draw (4 cm, 0.1 cm) -- +(2 cm,0);
\draw (4 cm, -0.1 cm) -- +(2 cm,0);
\draw[shift={(5.2, 0)}, rotate=0] (135 : 0.45cm) -- (0,0) -- (-135 : 0.45cm);
\draw[fill=white] (0 cm, 0 cm) circle (.25cm) node[below=4pt]{$1$};
\draw[fill=white] (2 cm, 0 cm) circle (.25cm) node[below=4pt]{$2$};
\draw[fill=white] (4 cm, 0 cm) circle (.25cm) node[below=4pt]{$3$};
\draw[fill=white] (6 cm, 0 cm) circle (.25cm) node[below=4pt]{$4$};
}
\draw[fill=white] (0 cm, 0 cm) circle (.25cm) node[below=4pt]{$0$};
<BLANKLINE>
"""
if node is None:
node = self._latex_draw_node
if self.n == 1:
import cartan_type
return cartan_type.CartanType(["A", 1, 1])._latex_dynkin_diagram(
label, node, node_dist)
ret = "\\draw (0, 0.1 cm) -- +(%s cm,0);\n" % node_dist
ret += "\\draw (0, -0.1 cm) -- +(%s cm,0);\n" % node_dist
if dual:
ret += self._latex_draw_arrow_tip(0.5 * node_dist - 0.2, 0, 180)
else:
ret += self._latex_draw_arrow_tip(0.5 * node_dist + 0.2, 0, 0)
ret += "{\n\\pgftransformxshift{%s cm}\n" % node_dist
ret += self.classical()._latex_dynkin_diagram(label, node, node_dist,
dual)
ret += "}\n" + node(0, 0, label(0))
return ret
