Coverage for local/lib/python2.7/site-packages/sage/combinat/root_system/type_D_affine.py : 94%

""" 

Root system data for (untwisted) type D affine 

""" 

#***************************************************************************** 

# Copyright (C) 2008-2009 Daniel Bump 

# Copyright (C) 2008-2009 Justin Walker 

# Copyright (C) 2008-2009 Nicolas M. Thiery <nthiery at users.sf.net>, 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

from __future__ import print_function 

from __future__ import absolute_import 

from .cartan_type import CartanType_standard_untwisted_affine, CartanType_simply_laced 

class CartanType(CartanType_standard_untwisted_affine, CartanType_simply_laced): 

def __init__(self, n): 

""" 

EXAMPLES:: 

sage: ct = CartanType(['D',4,1]) 

sage: ct 

['D', 4, 1] 

sage: ct._repr_(compact = True) 

'D4~' 

sage: ct.is_irreducible() 

True 

sage: ct.is_finite() 

False 

sage: ct.is_affine() 

True 

sage: ct.is_untwisted_affine() 

True 

sage: ct.is_crystallographic() 

True 

sage: ct.is_simply_laced() 

True 

sage: ct.classical() 

['D', 4] 

sage: ct.dual() 

['D', 4, 1] 

TESTS:: 

sage: TestSuite(ct).run() 

""" 

assert n >= 3 

CartanType_standard_untwisted_affine.__init__(self, "D", n) 

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: 

from . 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(['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 cm, 0.7 cm) circle (.25cm) node[left=3pt]{$0$}; 

\draw[fill=white] (0 cm, -0.7 cm) circle (.25cm) node[left=3pt]{$1$}; 

\draw[fill=white] (2 cm, 0 cm) 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]{$3$}; 

<BLANKLINE> 

""" 

if node is None: 

node = self._latex_draw_node 

n = self.n 

if n == 3: 

from . 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.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 += node(0, 0.7, label(0), "left=3pt") 

ret += node(0, -0.7, label(1), "left=3pt") 

for i in range(1, self.n-2): 

ret += node(i*node_dist, 0, label(i+1)) 

ret += node(rt_most, 0.7, label(n), "right=3pt") 

ret += node(rt_most, -0.7, label(n-1), "right=3pt") 

return ret 

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: 

from . 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