Root system data for type F

sage.combinat.root_system.type_F.AmbientSpace

The lattice behind \(F_4\). The computations are based on Bourbaki, Groupes et Algebres de Lie, Ch. 4,5,6 (planche VIII).

class sage.combinat.root_system.type_F.CartanType

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_crystallographic

EXAMPLES:

sage: ct = CartanType(['F',4])
sage: ct
['F', 4]
sage: ct._repr_(compact = True)
'F4'

sage: ct.is_irreducible()
True
sage: ct.is_finite()
True
sage: ct.is_crystallographic()
True
sage: ct.is_simply_laced()
False
sage: ct.dual()
['F', 4] relabelled by {1: 4, 2: 3, 3: 2, 4: 1}
sage: ct.affine()
['F', 4, 1]
AmbientSpace

The lattice behind \(F_4\). The computations are based on Bourbaki, Groupes et Algebres de Lie, Ch. 4,5,6 (planche VIII).

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

Return an ascii art representation of the extended Dynkin diagram.

EXAMPLES:

sage: print(CartanType(['F',4]).ascii_art(label = lambda x: x+2))
O---O=>=O---O
3   4   5   6
sage: print(CartanType(['F',4]).ascii_art(label = lambda x: x-2))
O---O=>=O---O
-1  0   1   2
coxeter_number()

Return the Coxeter number associated with self.

EXAMPLES:

sage: CartanType(['F',4]).coxeter_number()
12
dual()

Return the dual Cartan type.

This uses that \(F_4\) is self-dual up to relabelling.

EXAMPLES:

sage: F4 = CartanType(['F',4])
sage: F4.dual()
['F', 4] relabelled by {1: 4, 2: 3, 3: 2, 4: 1}

sage: F4.dynkin_diagram()
O---O=>=O---O
1   2   3   4
F4
sage: F4.dual().dynkin_diagram()
O---O=>=O---O
4   3   2   1
F4 relabelled by {1: 4, 2: 3, 3: 2, 4: 1}
dual_coxeter_number()

Return the dual Coxeter number associated with self.

EXAMPLES:

sage: CartanType(['F',4]).dual_coxeter_number()
9
dynkin_diagram()

Returns a Dynkin diagram for type F.

EXAMPLES:

sage: f = CartanType(['F',4]).dynkin_diagram()
sage: f
O---O=>=O---O
1   2   3   4
F4
sage: sorted(f.edges())
[(1, 2, 1), (2, 1, 1), (2, 3, 2), (3, 2, 1), (3, 4, 1), (4, 3, 1)]