Root system data for type E

sage.combinat.root_system.type_E.AmbientSpace

The lattice behind E6, E7, or E8. The computations are based on Bourbaki, Groupes et Algebres de Lie, Ch. 4,5,6 (planche V-VII).

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

Bases: sage.combinat.root_system.cartan_type.CartanType_standard_finite, sage.combinat.root_system.cartan_type.CartanType_simple, sage.combinat.root_system.cartan_type.CartanType_simply_laced

EXAMPLES:

sage: ct = CartanType(['E',6])
sage: ct
['E', 6]
sage: ct._repr_(compact = True)
'E6'
sage: ct.is_irreducible()
True
sage: ct.is_finite()
True
sage: ct.is_affine()
False
sage: ct.is_crystallographic()
True
sage: ct.is_simply_laced()
True
sage: ct.affine()
['E', 6, 1]
sage: ct.dual()
['E', 6]
AmbientSpace

The lattice behind E6, E7, or E8. The computations are based on Bourbaki, Groupes et Algebres de Lie, Ch. 4,5,6 (planche V-VII).

ascii_art(label=<function CartanType.<lambda>>, node=None)

Return a ascii art representation of the extended Dynkin diagram.

EXAMPLES:

sage: print(CartanType(['E',6]).ascii_art(label = lambda x: x+2))
        O 4
        |
        |
O---O---O---O---O
3   5   6   7   8
sage: print(CartanType(['E',7]).ascii_art(label = lambda x: x+2))
        O 4
        |
        |
O---O---O---O---O---O
3   5   6   7   8   9
sage: print(CartanType(['E',8]).ascii_art(label = lambda x: x+1))
        O 3
        |
        |
O---O---O---O---O---O---O
2   4   5   6   7   8   9
coxeter_number()

Return the Coxeter number associated with self.

EXAMPLES:

sage: CartanType(['E',6]).coxeter_number()
12
sage: CartanType(['E',7]).coxeter_number()
18
sage: CartanType(['E',8]).coxeter_number()
30
dual_coxeter_number()

Return the dual Coxeter number associated with self.

EXAMPLES:

sage: CartanType(['E',6]).dual_coxeter_number()
12
sage: CartanType(['E',7]).dual_coxeter_number()
18
sage: CartanType(['E',8]).dual_coxeter_number()
30
dynkin_diagram()

Returns a Dynkin diagram for type E.

EXAMPLES:

sage: e = CartanType(['E',6]).dynkin_diagram()
sage: e
        O 2
        |
        |
O---O---O---O---O
1   3   4   5   6
E6
sage: sorted(e.edges())
[(1, 3, 1), (2, 4, 1), (3, 1, 1), (3, 4, 1), (4, 2, 1), (4, 3, 1), (4, 5, 1), (5, 4, 1), (5, 6, 1), (6, 5, 1)]
sage: e = CartanType(['E',7]).dynkin_diagram()
sage: e
        O 2
        |
        |
O---O---O---O---O---O
1   3   4   5   6   7
E7
sage: sorted(e.edges())
[(1, 3, 1), (2, 4, 1), (3, 1, 1), (3, 4, 1), (4, 2, 1),
 (4, 3, 1), (4, 5, 1), (5, 4, 1), (5, 6, 1), (6, 5, 1),
 (6, 7, 1), (7, 6, 1)]
sage: e = CartanType(['E',8]).dynkin_diagram()
sage: e
        O 2
        |
        |
O---O---O---O---O---O---O
1   3   4   5   6   7   8
E8
sage: sorted(e.edges())
[(1, 3, 1), (2, 4, 1), (3, 1, 1), (3, 4, 1), (4, 2, 1),
 (4, 3, 1), (4, 5, 1), (5, 4, 1), (5, 6, 1), (6, 5, 1),
 (6, 7, 1), (7, 6, 1), (7, 8, 1), (8, 7, 1)]