Root system data for (untwisted) type D affine

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

Bases: sage.combinat.root_system.cartan_type.CartanType_standard_untwisted_affine, sage.combinat.root_system.cartan_type.CartanType_simply_laced

EXAMPLES:

sage: ct = CartanType(['D',4,1])
sage: ct
['D', 4, 1]
sage: ct._repr_(compact = True)
'D4~'

sage: ct.is_irreducible()
True
sage: ct.is_finite()
False
sage: ct.is_affine()
True
sage: ct.is_untwisted_affine()
True
sage: ct.is_crystallographic()
True
sage: ct.is_simply_laced()
True
sage: ct.classical()
['D', 4]
sage: ct.dual()
['D', 4, 1]
PieriFactors

The type D affine Pieri factors are realized as the order ideal (in Bruhat order) generated by the following elements:

  • cyclic rotations of the element with reduced word 234…(n-2)n(n-1)(n-2)…3210 such that 1 and 0 are always adjacent and (n-1) and n are always adjacent.
  • 123…(n-2)n(n-1)(n-2)…321
  • 023…(n-2)n(n-1)(n-2)…320
  • n(n-2)…2102…(n-2)n
  • (n-1)(n-2)…2102…(n-2)(n-1)

EXAMPLES:

sage: W = WeylGroup(['D',5,1])
sage: PF = W.pieri_factors()
sage: W.from_reduced_word([3,2,1,0]) in PF
True
sage: W.from_reduced_word([0,3,2,1]) in PF
False
sage: W.from_reduced_word([0,1,3,2]) in PF
True
sage: W.from_reduced_word([2,0,1,3]) in PF
True
sage: sorted([u.reduced_word() for u in PF.maximal_elements()], key=str)
[[0, 2, 3, 5, 4, 3, 2, 0], [1, 0, 2, 3, 5, 4, 3, 2], [1, 2, 3, 5, 4, 3, 2, 1],
 [2, 1, 0, 2, 3, 5, 4, 3], [2, 3, 5, 4, 3, 2, 1, 0], [3, 2, 1, 0, 2, 3, 5, 4],
 [3, 5, 4, 3, 2, 1, 0, 2], [4, 3, 2, 1, 0, 2, 3, 4], [5, 3, 2, 1, 0, 2, 3, 5],
 [5, 4, 3, 2, 1, 0, 2, 3]]
ascii_art(label=<function CartanType.<lambda>>, node=None)

Return an ascii art representation of the extended Dynkin diagram.

dynkin_diagram()

Returns the extended Dynkin diagram for affine type D.

EXAMPLES:

sage: d = CartanType(['D', 6, 1]).dynkin_diagram()
sage: d
   0 O       O 6
     |       |
     |       |
 O---O---O---O---O
 1   2   3   4   5
 D6~
sage: sorted(d.edges())
[(0, 2, 1), (1, 2, 1), (2, 0, 1), (2, 1, 1), (2, 3, 1),
 (3, 2, 1), (3, 4, 1), (4, 3, 1), (4, 5, 1), (4, 6, 1), (5, 4, 1), (6, 4, 1)]

sage: d = CartanType(['D', 4, 1]).dynkin_diagram()
sage: d
    O 4
    |
    |
O---O---O
1   |2  3
    |
    O 0
D4~
sage: sorted(d.edges())
[(0, 2, 1),
 (1, 2, 1),
 (2, 0, 1),
 (2, 1, 1),
 (2, 3, 1),
 (2, 4, 1),
 (3, 2, 1),
 (4, 2, 1)]

sage: d = CartanType(['D', 3, 1]).dynkin_diagram()
sage: d
0
O-------+
|       |
|       |
O---O---O
3   1   2
D3~
sage: sorted(d.edges())
[(0, 2, 1), (0, 3, 1), (1, 2, 1), (1, 3, 1), (2, 0, 1), (2, 1, 1), (3, 0, 1), (3, 1, 1)]