def basic_untwisted(self):
r"""
Return the basic untwisted Cartan type associated with this affine
Cartan type.
Given an affine type `X_n^{(r)}`, the basic untwisted type is `X_n`.
In other words, it is the classical Cartan type that is twisted to
obtain ``self``.
EXAMPLES::
sage: CartanType(['A', 7, 2]).basic_untwisted()
['A', 7]
sage: CartanType(['E', 6, 2]).basic_untwisted()
['E', 6]
sage: CartanType(['D', 4, 3]).basic_untwisted()
['D', 4]
"""
from . import cartan_type
if self.dual().type() == 'B':
return cartan_type.CartanType(
['A', self.classical().rank() * 2 - 1])
elif self.dual().type() == 'BC':
return cartan_type.CartanType(['A', self.classical().rank() * 2])
elif self.dual().type() == 'C':
return cartan_type.CartanType(['D', self.classical().rank() + 1])
elif self.dual().type() == 'F':
return cartan_type.CartanType(['E', 6])
elif self.dual().type() == 'G':
return cartan_type.CartanType(['D', 4])
def __classcall__(cls, ct, marked_nodes):
"""
This standardizes the input of the constructor to ensure
unique representation.
EXAMPLES::
sage: ct1 = CartanType(['B',2]).marked_nodes([1,2])
sage: ct2 = CartanType(['B',2]).marked_nodes([2,1])
sage: ct3 = CartanType(['B',2]).marked_nodes((1,2))
sage: ct4 = CartanType(['D',4]).marked_nodes([1,2])
sage: ct1 is ct2
True
sage: ct1 is ct3
True
sage: ct1 == ct4
False
"""
ct = cartan_type.CartanType(ct)
if not marked_nodes:
return ct
if any(node not in ct.index_set() for node in marked_nodes):
raise ValueError("invalid marked node")
marked_nodes = tuple(sorted(marked_nodes))
return super(CartanType, cls).__classcall__(cls, ct, marked_nodes)
