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)
simple_root(i)[source]

EXAMPLES:

sage: RootSystem(['C',3]).ambient_space().simple_roots()
Finite family {1: (1, -1, 0), 2: (0, 1, -1), 3: (0, 0, 2)}

>>> from sage.all import *
>>> RootSystem(['C',Integer(3)]).ambient_space().simple_roots()
Finite family {1: (1, -1, 0), 2: (0, 1, -1), 3: (0, 0, 2)}
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
 []