Root system data for (untwisted) type D affine¶
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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]
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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]]
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ascii_art
(label=<function CartanType.<lambda>>, node=None)¶ Return an ascii art representation of the extended Dynkin diagram.
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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)]
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