Root system data for Cartan types with marked nodes

sage.combinat.root_system.type_marked.AmbientSpace

Ambient space for a marked finite Cartan type.

It is constructed in the canonical way from the ambient space of the original Cartan type.

EXAMPLES:

sage: L = CartanType(["F",4]).marked_nodes([1,3]).root_system().ambient_space(); L
Ambient space of the Root system of type ['F', 4] with nodes (1, 3) marked
sage: TestSuite(L).run()
sage.combinat.root_system.type_marked.CartanType

A class for Cartan types with marked nodes.

INPUT:

  • ct – a Cartan type
  • marked_nodes – a list of marked nodes

EXAMPLES:

We take the Cartan type \(B_4\):

sage: T = CartanType(['B',4])
sage: T.dynkin_diagram()
O---O---O=>=O
1   2   3   4
B4

And mark some of its nodes:

sage: T = T.marked_nodes([2,3])
sage: T.dynkin_diagram()
O---X---X=>=O
1   2   3   4
B4 with nodes (2, 3) marked

Markings are not additive:

sage: T.marked_nodes([1,4]).dynkin_diagram()
X---O---O=>=X
1   2   3   4
B4 with nodes (1, 4) marked

And trivial relabelling are honoured nicely:

sage: T = T.marked_nodes([])
sage: T.dynkin_diagram()
O---O---O=>=O
1   2   3   4
B4
sage.combinat.root_system.type_marked.CartanType_affine
class sage.combinat.root_system.type_marked.CartanType_finite(ct, marked_nodes)

Bases: sage.combinat.root_system.type_marked.CartanType, sage.combinat.root_system.cartan_type.CartanType_finite

AmbientSpace

Ambient space for a marked finite Cartan type.

It is constructed in the canonical way from the ambient space of the original Cartan type.

EXAMPLES:

sage: L = CartanType(["F",4]).marked_nodes([1,3]).root_system().ambient_space(); L
Ambient space of the Root system of type ['F', 4] with nodes (1, 3) marked
sage: TestSuite(L).run()
affine()

Return the affine Cartan type associated with self.

EXAMPLES:

sage: B4 = CartanType(['B',4]).marked_nodes([1,3])
sage: B4.dynkin_diagram()
X---O---X=>=O
1   2   3   4
B4 with nodes (1, 3) marked
sage: B4.affine().dynkin_diagram()
    O 0
    |
    |
X---O---X=>=O
1   2   3   4
B4~ with nodes (1, 3) marked