Root system data for type D#
- class sage.combinat.root_system.type_D.AmbientSpace(root_system, base_ring, index_set=None)#
Bases:
AmbientSpace- dimension()#
EXAMPLES:
sage: e = RootSystem(['D',3]).ambient_space() sage: e.dimension() 3
- fundamental_weight(i)#
EXAMPLES:
sage: RootSystem(['D',4]).ambient_space().fundamental_weights() Finite family {1: (1, 0, 0, 0), 2: (1, 1, 0, 0), 3: (1/2, 1/2, 1/2, -1/2), 4: (1/2, 1/2, 1/2, 1/2)}
- negative_roots()#
EXAMPLES:
sage: RootSystem(['D',4]).ambient_space().negative_roots() [(-1, 1, 0, 0), (-1, 0, 1, 0), (0, -1, 1, 0), (-1, 0, 0, 1), (0, -1, 0, 1), (0, 0, -1, 1), (-1, -1, 0, 0), (-1, 0, -1, 0), (0, -1, -1, 0), (-1, 0, 0, -1), (0, -1, 0, -1), (0, 0, -1, -1)]
- positive_roots()#
EXAMPLES:
sage: RootSystem(['D',4]).ambient_space().positive_roots() [(1, 1, 0, 0), (1, 0, 1, 0), (0, 1, 1, 0), (1, 0, 0, 1), (0, 1, 0, 1), (0, 0, 1, 1), (1, -1, 0, 0), (1, 0, -1, 0), (0, 1, -1, 0), (1, 0, 0, -1), (0, 1, 0, -1), (0, 0, 1, -1)]
- root(i, j, p1, p2)#
Note that indexing starts at 0.
EXAMPLES:
sage: e = RootSystem(['D',3]).ambient_space() sage: e.root(0, 1, 1, 1) (-1, -1, 0) sage: e.root(0, 0, 1, 1) (-1, 0, 0)
- simple_root(i)#
EXAMPLES:
sage: RootSystem(['D',4]).ambient_space().simple_roots() Finite family {1: (1, -1, 0, 0), 2: (0, 1, -1, 0), 3: (0, 0, 1, -1), 4: (0, 0, 1, 1)}
- class sage.combinat.root_system.type_D.CartanType(n)#
Bases:
CartanType_standard_finite,CartanType_simply_lacedEXAMPLES:
sage: ct = CartanType(['D',4]) sage: ct ['D', 4] sage: ct._repr_(compact = True) 'D4' sage: ct.is_irreducible() True sage: ct.is_finite() True sage: ct.is_crystallographic() True sage: ct.is_simply_laced() True sage: ct.dual() ['D', 4] sage: ct.affine() ['D', 4, 1] sage: ct = CartanType(['D',2]) sage: ct.is_irreducible() False sage: ct.dual() ['D', 2] sage: ct.affine() Traceback (most recent call last): ... ValueError: ['D', 2, 1] is not a valid Cartan type
- AmbientSpace#
alias of
AmbientSpace
- ascii_art(label=None, node=None)#
Return a ascii art representation of the extended Dynkin diagram.
EXAMPLES:
sage: print(CartanType(['D',3]).ascii_art()) O 3 | | O---O 1 2 sage: print(CartanType(['D',4]).ascii_art()) O 4 | | O---O---O 1 2 3 sage: print(CartanType(['D',4]).ascii_art(label = lambda x: x+2)) O 6 | | O---O---O 3 4 5 sage: print(CartanType(['D',6]).ascii_art(label = lambda x: x+2)) O 8 | | O---O---O---O---O 3 4 5 6 7
- coxeter_number()#
Return the Coxeter number associated with
self.EXAMPLES:
sage: CartanType(['D',4]).coxeter_number() 6
- dual_coxeter_number()#
Return the dual Coxeter number associated with
self.EXAMPLES:
sage: CartanType(['D',4]).dual_coxeter_number() 6
- dynkin_diagram()#
Returns a Dynkin diagram for type D.
EXAMPLES:
sage: d = CartanType(['D',5]).dynkin_diagram(); d # needs sage.graphs O 5 | | O---O---O---O 1 2 3 4 D5 sage: d.edges(sort=True) # needs sage.graphs [(1, 2, 1), (2, 1, 1), (2, 3, 1), (3, 2, 1), (3, 4, 1), (3, 5, 1), (4, 3, 1), (5, 3, 1)] sage: d = CartanType(['D',4]).dynkin_diagram(); d # needs sage.graphs O 4 | | O---O---O 1 2 3 D4 sage: d.edges(sort=True) # needs sage.graphs [(1, 2, 1), (2, 1, 1), (2, 3, 1), (2, 4, 1), (3, 2, 1), (4, 2, 1)] sage: d = CartanType(['D',3]).dynkin_diagram(); d # needs sage.graphs O 3 | | O---O 1 2 D3 sage: d.edges(sort=True) # needs sage.graphs [(1, 2, 1), (1, 3, 1), (2, 1, 1), (3, 1, 1)] sage: d = CartanType(['D',2]).dynkin_diagram(); d # needs sage.graphs O O 1 2 D2 sage: d.edges(sort=True) # needs sage.graphs []
- is_atomic()#
Implements
CartanType_abstract.is_atomic()\(D_2\) is atomic, like all \(D_n\), despite being non irreducible.
EXAMPLES:
sage: CartanType(["D",2]).is_atomic() True sage: CartanType(["D",2]).is_irreducible() False