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""" Root system data for (untwisted) type E 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/ #*****************************************************************************
""" EXAMPLES::
sage: ct = CartanType(['E',6,1]) sage: ct ['E', 6, 1] sage: ct._repr_(compact = True) 'E6~'
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() ['E', 6] sage: ct.dual() ['E', 6, 1]
TESTS::
sage: TestSuite(ct).run() """ raise ValueError("Invalid Cartan Type for Type E")
""" Return a latex representation of ``self``.
EXAMPLES::
sage: latex(CartanType(['E',7,1])) E_7^{(1)} """
""" 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)]
""" else: raise ValueError("Invalid Cartan Type for Type E affine")
r""" Return a latex representation of the Dynkin diagram.
EXAMPLES::
sage: print(CartanType(['E',7,1])._latex_dynkin_diagram()) \draw (0 cm,0) -- (12 cm,0); \draw (6 cm, 0 cm) -- +(0,2 cm); \draw[fill=white] (0 cm, 0 cm) circle (.25cm) node[below=4pt]{$0$}; \draw[fill=white] (2 cm, 0 cm) circle (.25cm) node[below=4pt]{$1$}; \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] (8 cm, 0 cm) circle (.25cm) node[below=4pt]{$5$}; \draw[fill=white] (10 cm, 0 cm) circle (.25cm) node[below=4pt]{$6$}; \draw[fill=white] (12 cm, 0 cm) circle (.25cm) node[below=4pt]{$7$}; \draw[fill=white] (6 cm, 2 cm) circle (.25cm) node[right=3pt]{$2$}; <BLANKLINE> """
ret = "\\draw (0 cm,0) -- (%s cm,0);\n"%((n-2)*node_dist) ret += "\\draw (%s cm, 0 cm) -- +(0,%s cm);\n"%(2*node_dist, node_dist)
if n == 6: ret += "\\draw (%s cm, %s cm) -- +(0,%s cm);\n"%(2*node_dist, node_dist, node_dist) ret += node(2*node_dist, 2*node_dist, label(0), "right=3pt") else: # n == 8 ret += "\\draw (%s cm,0) -- +(%s cm,0);\n"%((n-2)*node_dist, node_dist) ret += node((n-1)*node_dist, 0, label(0))
ret += node(0, 0, label(1)) for i in range(1, n-1): ret += node(i*node_dist, 0, label(i+2)) ret += node(2*node_dist, node_dist, label(2), "right=3pt") return ret
""" Return an ascii art representation of the extended Dynkin diagram.
EXAMPLES::
sage: print(CartanType(['E',6,1]).ascii_art(label = lambda x: x+2)) O 2 | | O 4 | | O---O---O---O---O 3 5 6 7 8 sage: print(CartanType(['E',7,1]).ascii_art(label = lambda x: x+2)) O 4 | | O---O---O---O---O---O---O 2 3 5 6 7 8 9 sage: print(CartanType(['E',8,1]).ascii_art(label = lambda x: x-3)) O -1 | | O---O---O---O---O---O---O---O -2 0 1 2 3 4 5 -3 """
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