Root system data for type C¶
- class sage.combinat.root_system.type_C.AmbientSpace(root_system, base_ring, index_set=None)[source]¶
- Bases: - AmbientSpace- EXAMPLES: - sage: e = RootSystem(['C',2]).ambient_space(); e Ambient space of the Root system of type ['C', 2] - >>> from sage.all import * >>> e = RootSystem(['C',Integer(2)]).ambient_space(); e Ambient space of the Root system of type ['C', 2] - One cannot construct the ambient lattice because the fundamental coweights have rational coefficients: - sage: e.smallest_base_ring() Rational Field sage: RootSystem(['B',2]).ambient_space().fundamental_weights() Finite family {1: (1, 0), 2: (1/2, 1/2)} - >>> from sage.all import * >>> e.smallest_base_ring() Rational Field >>> RootSystem(['B',Integer(2)]).ambient_space().fundamental_weights() Finite family {1: (1, 0), 2: (1/2, 1/2)} - dimension()[source]¶
- EXAMPLES: - sage: e = RootSystem(['C',3]).ambient_space() sage: e.dimension() 3 - >>> from sage.all import * >>> e = RootSystem(['C',Integer(3)]).ambient_space() >>> e.dimension() 3 
 - fundamental_weight(i)[source]¶
- EXAMPLES: - sage: RootSystem(['C',3]).ambient_space().fundamental_weights() Finite family {1: (1, 0, 0), 2: (1, 1, 0), 3: (1, 1, 1)} - >>> from sage.all import * >>> RootSystem(['C',Integer(3)]).ambient_space().fundamental_weights() Finite family {1: (1, 0, 0), 2: (1, 1, 0), 3: (1, 1, 1)} 
 - negative_roots()[source]¶
- EXAMPLES: - sage: RootSystem(['C',3]).ambient_space().negative_roots() [(-1, 1, 0), (-1, 0, 1), (0, -1, 1), (-1, -1, 0), (-1, 0, -1), (0, -1, -1), (-2, 0, 0), (0, -2, 0), (0, 0, -2)] - >>> from sage.all import * >>> RootSystem(['C',Integer(3)]).ambient_space().negative_roots() [(-1, 1, 0), (-1, 0, 1), (0, -1, 1), (-1, -1, 0), (-1, 0, -1), (0, -1, -1), (-2, 0, 0), (0, -2, 0), (0, 0, -2)] 
 - positive_roots()[source]¶
- EXAMPLES: - sage: RootSystem(['C',3]).ambient_space().positive_roots() [(1, 1, 0), (1, 0, 1), (0, 1, 1), (1, -1, 0), (1, 0, -1), (0, 1, -1), (2, 0, 0), (0, 2, 0), (0, 0, 2)] - >>> from sage.all import * >>> RootSystem(['C',Integer(3)]).ambient_space().positive_roots() [(1, 1, 0), (1, 0, 1), (0, 1, 1), (1, -1, 0), (1, 0, -1), (0, 1, -1), (2, 0, 0), (0, 2, 0), (0, 0, 2)] 
 - root(i, j, p1, p2)[source]¶
- Note that indexing starts at 0. - EXAMPLES: - sage: e = RootSystem(['C',3]).ambient_space() sage: e.root(0, 1, 1, 1) (-1, -1, 0) - >>> from sage.all import * >>> e = RootSystem(['C',Integer(3)]).ambient_space() >>> e.root(Integer(0), Integer(1), Integer(1), Integer(1)) (-1, -1, 0) 
 
- class sage.combinat.root_system.type_C.CartanType(n)[source]¶
- Bases: - CartanType_standard_finite,- CartanType_simple,- CartanType_crystallographic- EXAMPLES: - sage: ct = CartanType(['C',4]) sage: ct ['C', 4] sage: ct._repr_(compact = True) 'C4' sage: ct.is_irreducible() True sage: ct.is_finite() True sage: ct.is_crystallographic() True sage: ct.is_simply_laced() False sage: ct.affine() ['C', 4, 1] sage: ct.dual() ['B', 4] sage: ct = CartanType(['C',1]) sage: ct.is_simply_laced() True sage: ct.affine() ['C', 1, 1] - >>> from sage.all import * >>> ct = CartanType(['C',Integer(4)]) >>> ct ['C', 4] >>> ct._repr_(compact = True) 'C4' >>> ct.is_irreducible() True >>> ct.is_finite() True >>> ct.is_crystallographic() True >>> ct.is_simply_laced() False >>> ct.affine() ['C', 4, 1] >>> ct.dual() ['B', 4] >>> ct = CartanType(['C',Integer(1)]) >>> ct.is_simply_laced() True >>> ct.affine() ['C', 1, 1] - AmbientSpace[source]¶
- alias of - AmbientSpace
 - ascii_art(label=None, node=None)[source]¶
- Return a ascii art representation of the extended Dynkin diagram. - EXAMPLES: - sage: print(CartanType(['C',1]).ascii_art()) O 1 sage: print(CartanType(['C',2]).ascii_art()) O=<=O 1 2 sage: print(CartanType(['C',3]).ascii_art()) O---O=<=O 1 2 3 sage: print(CartanType(['C',5]).ascii_art(label = lambda x: x+2)) O---O---O---O=<=O 3 4 5 6 7 - >>> from sage.all import * >>> print(CartanType(['C',Integer(1)]).ascii_art()) O 1 >>> print(CartanType(['C',Integer(2)]).ascii_art()) O=<=O 1 2 >>> print(CartanType(['C',Integer(3)]).ascii_art()) O---O=<=O 1 2 3 >>> print(CartanType(['C',Integer(5)]).ascii_art(label = lambda x: x+Integer(2))) O---O---O---O=<=O 3 4 5 6 7 
 - coxeter_number()[source]¶
- Return the Coxeter number associated with - self.- EXAMPLES: - sage: CartanType(['C',4]).coxeter_number() 8 - >>> from sage.all import * >>> CartanType(['C',Integer(4)]).coxeter_number() 8 
 - dual()[source]¶
- Types B and C are in duality: - EXAMPLES: - sage: CartanType(["C", 3]).dual() ['B', 3] - >>> from sage.all import * >>> CartanType(["C", Integer(3)]).dual() ['B', 3] 
 - dual_coxeter_number()[source]¶
- Return the dual Coxeter number associated with - self.- EXAMPLES: - sage: CartanType(['C',4]).dual_coxeter_number() 5 - >>> from sage.all import * >>> CartanType(['C',Integer(4)]).dual_coxeter_number() 5 
 - dynkin_diagram()[source]¶
- Return a Dynkin diagram for type C. - EXAMPLES: - sage: c = CartanType(['C',3]).dynkin_diagram(); c # needs sage.graphs O---O=<=O 1 2 3 C3 sage: c.edges(sort=True) # needs sage.graphs [(1, 2, 1), (2, 1, 1), (2, 3, 1), (3, 2, 2)] sage: b = CartanType(['C',1]).dynkin_diagram(); b # needs sage.graphs O 1 C1 sage: b.edges(sort=True) # needs sage.graphs [] - >>> from sage.all import * >>> c = CartanType(['C',Integer(3)]).dynkin_diagram(); c # needs sage.graphs O---O=<=O 1 2 3 C3 >>> c.edges(sort=True) # needs sage.graphs [(1, 2, 1), (2, 1, 1), (2, 3, 1), (3, 2, 2)] >>> b = CartanType(['C',Integer(1)]).dynkin_diagram(); b # needs sage.graphs O 1 C1 >>> b.edges(sort=True) # needs sage.graphs []