Root system data for (untwisted) type A affine¶
- class sage.combinat.root_system.type_A_affine.CartanType(n)[source]¶
- Bases: - CartanType_standard_untwisted_affine- EXAMPLES: - sage: ct = CartanType(['A',4,1]) sage: ct ['A', 4, 1] sage: ct._repr_(compact = True) 'A4~' 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() ['A', 4] sage: ct.dual() ['A', 4, 1] sage: ct = CartanType(['A', 1, 1]) sage: ct.is_simply_laced() False sage: ct.dual() ['A', 1, 1] - >>> from sage.all import * >>> ct = CartanType(['A',Integer(4),Integer(1)]) >>> ct ['A', 4, 1] >>> ct._repr_(compact = True) 'A4~' >>> 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() ['A', 4] >>> ct.dual() ['A', 4, 1] >>> ct = CartanType(['A', Integer(1), Integer(1)]) >>> ct.is_simply_laced() False >>> ct.dual() ['A', 1, 1] - PieriFactors[source]¶
- alias of - PieriFactors_type_A_affine
 - ascii_art(label=None, node=None)[source]¶
- Return an ascii art representation of the extended Dynkin diagram. - EXAMPLES: - sage: print(CartanType(['A',3,1]).ascii_art()) 0 O-------+ | | | | O---O---O 1 2 3 sage: print(CartanType(['A',5,1]).ascii_art(label = lambda x: x+2)) 2 O---------------+ | | | | O---O---O---O---O 3 4 5 6 7 sage: print(CartanType(['A',1,1]).ascii_art()) O<=>O 0 1 sage: print(CartanType(['A',1,1]).ascii_art(label = lambda x: x+2)) O<=>O 2 3 - >>> from sage.all import * >>> print(CartanType(['A',Integer(3),Integer(1)]).ascii_art()) 0 O-------+ | | | | O---O---O 1 2 3 >>> print(CartanType(['A',Integer(5),Integer(1)]).ascii_art(label = lambda x: x+Integer(2))) 2 O---------------+ | | | | O---O---O---O---O 3 4 5 6 7 >>> print(CartanType(['A',Integer(1),Integer(1)]).ascii_art()) O<=>O 0 1 >>> print(CartanType(['A',Integer(1),Integer(1)]).ascii_art(label = lambda x: x+Integer(2))) O<=>O 2 3 
 - dual()[source]¶
- Type \(A_1^1\) is self dual despite not being simply laced. - EXAMPLES: - sage: CartanType(['A',1,1]).dual() ['A', 1, 1] - >>> from sage.all import * >>> CartanType(['A',Integer(1),Integer(1)]).dual() ['A', 1, 1] 
 - dynkin_diagram()[source]¶
- Return the extended Dynkin diagram for affine type A. - EXAMPLES: - sage: a = CartanType(['A',3,1]).dynkin_diagram(); a # needs sage.graphs 0 O-------+ | | | | O---O---O 1 2 3 A3~ sage: a.edges(sort=True) # needs sage.graphs [(0, 1, 1), (0, 3, 1), (1, 0, 1), (1, 2, 1), (2, 1, 1), (2, 3, 1), (3, 0, 1), (3, 2, 1)] sage: a = DynkinDiagram(['A',1,1]); a # needs sage.graphs O<=>O 0 1 A1~ sage: a.edges(sort=True) # needs sage.graphs [(0, 1, 2), (1, 0, 2)] - >>> from sage.all import * >>> a = CartanType(['A',Integer(3),Integer(1)]).dynkin_diagram(); a # needs sage.graphs 0 O-------+ | | | | O---O---O 1 2 3 A3~ >>> a.edges(sort=True) # needs sage.graphs [(0, 1, 1), (0, 3, 1), (1, 0, 1), (1, 2, 1), (2, 1, 1), (2, 3, 1), (3, 0, 1), (3, 2, 1)] >>> a = DynkinDiagram(['A',Integer(1),Integer(1)]); a # needs sage.graphs O<=>O 0 1 A1~ >>> a.edges(sort=True) # needs sage.graphs [(0, 1, 2), (1, 0, 2)]