Root system data for (untwisted) type E affine#
- class sage.combinat.root_system.type_E_affine.CartanType(n)[source]#
Bases:
CartanType_standard_untwisted_affine,CartanType_simply_lacedEXAMPLES:
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]
>>> from sage.all import * >>> ct = CartanType(['E',Integer(6),Integer(1)]) >>> ct ['E', 6, 1] >>> ct._repr_(compact = True) 'E6~' >>> ct.is_irreducible() True >>> ct.is_finite() False >>> ct.is_affine() True >>> ct.is_untwisted_affine() True >>> ct.is_crystallographic() True >>> ct.is_simply_laced() True >>> ct.classical() ['E', 6] >>> ct.dual() ['E', 6, 1]
- ascii_art(label=None, node=None)[source]#
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
>>> from sage.all import * >>> print(CartanType(['E',Integer(6),Integer(1)]).ascii_art(label = lambda x: x+Integer(2))) O 2 | | O 4 | | O---O---O---O---O 3 5 6 7 8 >>> print(CartanType(['E',Integer(7),Integer(1)]).ascii_art(label = lambda x: x+Integer(2))) O 4 | | O---O---O---O---O---O---O 2 3 5 6 7 8 9 >>> print(CartanType(['E',Integer(8),Integer(1)]).ascii_art(label = lambda x: x-Integer(3))) O -1 | | O---O---O---O---O---O---O---O -2 0 1 2 3 4 5 -3
- dynkin_diagram()[source]#
Returns the extended Dynkin diagram for affine type E.
EXAMPLES:
sage: e = CartanType(['E', 6, 1]).dynkin_diagram(); e # needs sage.graphs O 0 | | O 2 | | O---O---O---O---O 1 3 4 5 6 E6~ sage: e.edges(sort=True) # needs sage.graphs [(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: # needs sage.graphs sage: e = CartanType(['E', 7, 1]).dynkin_diagram(); e O 2 | | O---O---O---O---O---O---O 0 1 3 4 5 6 7 E7~ sage: e.edges(sort=True) [(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(); e O 2 | | O---O---O---O---O---O---O---O 1 3 4 5 6 7 8 0 E8~ sage: e.edges(sort=True) [(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 sage.all import * >>> e = CartanType(['E', Integer(6), Integer(1)]).dynkin_diagram(); e # needs sage.graphs O 0 | | O 2 | | O---O---O---O---O 1 3 4 5 6 E6~ >>> e.edges(sort=True) # needs sage.graphs [(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)] >>> # needs sage.graphs >>> e = CartanType(['E', Integer(7), Integer(1)]).dynkin_diagram(); e O 2 | | O---O---O---O---O---O---O 0 1 3 4 5 6 7 E7~ >>> e.edges(sort=True) [(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)] >>> e = CartanType(['E', Integer(8), Integer(1)]).dynkin_diagram(); e O 2 | | O---O---O---O---O---O---O---O 1 3 4 5 6 7 8 0 E8~ >>> e.edges(sort=True) [(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)]