Root system data for type I

class sage.combinat.root_system.type_I.CartanType(n)

Bases: CartanType_standard_finite, CartanType_simple

EXAMPLES:

Sage

sage: ct = CartanType(['I',5])
sage: ct
['I', 5]
sage: ct._repr_(compact = True)
'I5'
sage: ct.rank()
2
sage: ct.index_set()
(1, 2)

sage: ct.is_irreducible()
True
sage: ct.is_finite()
True
sage: ct.is_affine()
False
sage: ct.is_crystallographic()
False
sage: ct.is_simply_laced()
False

Python

>>> from sage.all import *
>>> ct = CartanType(['I',Integer(5)])
>>> ct
['I', 5]
>>> ct._repr_(compact = True)
'I5'
>>> ct.rank()
2
>>> ct.index_set()
(1, 2)

>>> ct.is_irreducible()
True
>>> ct.is_finite()
True
>>> ct.is_affine()
False
>>> ct.is_crystallographic()
False
>>> ct.is_simply_laced()
False

coxeter_diagram()

Returns the Coxeter matrix for this type.

EXAMPLES:

Sage

sage: ct = CartanType(['I', 4])
sage: ct.coxeter_diagram()                                                  # needs sage.graphs
Graph on 2 vertices
sage: ct.coxeter_diagram().edges(sort=True)                                 # needs sage.graphs
[(1, 2, 4)]
sage: ct.coxeter_matrix()                                                   # needs sage.graphs
[1 4]
[4 1]

Python

>>> from sage.all import *
>>> ct = CartanType(['I', Integer(4)])
>>> ct.coxeter_diagram()                                                  # needs sage.graphs
Graph on 2 vertices
>>> ct.coxeter_diagram().edges(sort=True)                                 # needs sage.graphs
[(1, 2, 4)]
>>> ct.coxeter_matrix()                                                   # needs sage.graphs
[1 4]
[4 1]

coxeter_number()

Return the Coxeter number associated with self.

EXAMPLES:

Sage

sage: CartanType(['I',3]).coxeter_number()
3
sage: CartanType(['I',12]).coxeter_number()
12

Python

>>> from sage.all import *
>>> CartanType(['I',Integer(3)]).coxeter_number()
3
>>> CartanType(['I',Integer(12)]).coxeter_number()
12

index_set()

Type \(I_2(p)\) is indexed by \(\{1,2\}\).

EXAMPLES:

Sage

sage: CartanType(['I', 5]).index_set()
(1, 2)

Python

>>> from sage.all import *
>>> CartanType(['I', Integer(5)]).index_set()
(1, 2)

rank()

Type \(I_2(p)\) is of rank 2.

EXAMPLES:

Sage

sage: CartanType(['I', 5]).rank()
2

Python

>>> from sage.all import *
>>> CartanType(['I', Integer(5)]).rank()
2