Coverage for local/lib/python2.7/site-packages/sage/combinat/root_system/type_BC_affine.py : 86%

""" 

Root system data for type BC 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_affine 

from sage.rings.integer_ring import ZZ 

class CartanType(CartanType_standard_affine): 

def __init__(self, n): 

""" 

EXAMPLES:: 

sage: ct = CartanType(['BC',4,2]) 

sage: ct 

['BC', 4, 2] 

sage: ct._repr_(compact = True) 

'BC4~' 

sage: ct.dynkin_diagram() 

O=<=O---O---O=<=O 

0 1 2 3 4 

BC4~ 

sage: ct.is_irreducible() 

True 

sage: ct.is_finite() 

False 

sage: ct.is_affine() 

True 

sage: ct.is_crystallographic() 

True 

sage: ct.is_simply_laced() 

False 

sage: ct.classical() 

['C', 4] 

sage: dual = ct.dual() 

sage: dual.dynkin_diagram() 

O=>=O---O---O=>=O 

0 1 2 3 4 

BC4~* 

sage: dual.special_node() 

0 

sage: dual.classical().dynkin_diagram() 

O---O---O=>=O 

1 2 3 4 

B4 

sage: CartanType(['BC',1,2]).dynkin_diagram() 

4 

O=<=O 

0 1 

BC1~ 

TESTS:: 

sage: TestSuite(ct).run() 

""" 

assert n in ZZ and n >= 1 

CartanType_standard_affine.__init__(self, "BC", n, 2) 

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 _latex_(self): 

r""" 

Return a latex representation of ``self``. 

EXAMPLES:: 

sage: latex(CartanType(['BC',4,2])) 

BC_{4}^{(2)} 

sage: CartanType.options.notation = 'Kac' 

sage: latex(CartanType(['BC',4,2])) 

A_{8}^{(2)} 

sage: latex(CartanType(['A',8,2])) 

A_{8}^{(2)} 

sage: CartanType.options._reset() 

""" 

if self.options.notation == "Kac": 

return "A_{%s}^{(2)}"%(2*self.classical().rank()) 

else: 

return "BC_{%s}^{(2)}"%self.n 

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(['BC',4,2])._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={(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]{$0$}; 

<BLANKLINE> 

sage: print(CartanType(['BC',4,2]).dual()._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={(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: 

ret = "\\draw (0, 0.05 cm) -- +(%s cm,0);\n"%node_dist 

ret += "\\draw (0, -0.05 cm) -- +(%s cm,0);\n"%node_dist 

ret += "\\draw (0, 0.15 cm) -- +(%s cm,0);\n"%node_dist 

ret += "\\draw (0, -0.15 cm) -- +(%s cm,0);\n"%node_dist 

if dual: 

ret += self._latex_draw_arrow_tip(0.5*node_dist+0.2, 0, 0) 

else: 

ret += self._latex_draw_arrow_tip(0.5*node_dist-0.2, 0, 180) 

ret += node(0, 0, label(0)) 

ret += node(node_dist, 0, label(1)) 

return ret 

n = self.n 

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, 0) 

else: 

ret += self._latex_draw_arrow_tip(0.5*node_dist-0.2, 0, 180) 

ret += "{\n\\pgftransformxshift{%s cm}\n"%node_dist 

classical = self.classical() 

ret += self.classical()._latex_dynkin_diagram(label, node, node_dist, dual=dual) 

ret += "}\n" + node(0, 0, label(0)) 

return ret 

def ascii_art(self, label=lambda i: i, node=None): 

""" 

Return a ascii art representation of the extended Dynkin diagram. 

EXAMPLES:: 

sage: print(CartanType(['BC',2,2]).ascii_art()) 

O=<=O=<=O 

0 1 2 

sage: print(CartanType(['BC',3,2]).ascii_art()) 

O=<=O---O=<=O 

0 1 2 3 

sage: print(CartanType(['BC',5,2]).ascii_art(label = lambda x: x+2)) 

O=<=O---O---O---O=<=O 

2 3 4 5 6 7 

sage: print(CartanType(['BC',1,2]).ascii_art(label = lambda x: x+2)) 

4 

O=<=O 

2 3 

""" 

if node is None: 

node = self._ascii_art_node 

n = self.n 

if n == 1: 

return " 4\n{}=<={}\n{!s:4}{!s:4}".format(node(label(0)), node(label(1)), label(0), label(1)) 

ret = node(label(0)) + "=<=" + "---".join(node(label(i)) for i in range(1,n)) 

ret += "=<=" + node(label(n)) + '\n' 

ret += "".join("{!s:4}".format(label(i)) for i in range(n+1)) 

return ret 

def classical(self): 

""" 

Returns the classical Cartan type associated with self 

sage: CartanType(["BC", 3, 2]).classical() 

['C', 3] 

""" 

from . import cartan_type 

return cartan_type.CartanType(["C", self.n]) 

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] 

""" 

from . import cartan_type 

return cartan_type.CartanType(["A", 2*self.n]) 

def _default_folded_cartan_type(self): 

""" 

Return the default folded Cartan type. 

EXAMPLES:: 

sage: CartanType(['BC', 3, 2])._default_folded_cartan_type() 

['BC', 3, 2] as a folding of ['A', 5, 1] 

""" 

from sage.combinat.root_system.type_folded import CartanTypeFolded 

n = self.n 

return CartanTypeFolded(self, ['A', 2*n - 1, 1], 

[[0]] + [[i, 2*n-i] for i in range(1, n)] + [[n]])