Root system data for (untwisted) type A affine

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

Bases: sage.combinat.root_system.cartan_type.CartanType_standard_untwisted_affine

EXAMPLES:

sage: ct = CartanType(['A',4,1])
sage: ct
['A', 4, 1]
sage: ct._repr_(compact = True)
'A4~'

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()
['A', 4]
sage: ct.dual()
['A', 4, 1]

sage: ct = CartanType(['A', 1, 1])
sage: ct.is_simply_laced()
False
sage: ct.dual()
['A', 1, 1]
PieriFactors

The set of Pieri factors for type A affine, that is the set of elements of the Weyl Group which are cyclically decreasing.

Those are used for constructing (affine) Stanley symmetric functions.

The Pieri factors are in bijection with the proper subsets of the index_set. The bijection is given by the support. Namely, let \(f\) be a Pieri factor, and \(red\) a reduced word for \(f\). No simple reflection appears twice in red, and the support \(S\) of \(red\) (that is the \(i\) such that \(s_i\) appears in \(red\)) does not depend on the reduced word).

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

Return an ascii art representation of the extended Dynkin diagram.

EXAMPLES:

sage: print(CartanType(['A',3,1]).ascii_art())
0
O-------+
|       |
|       |
O---O---O
1   2   3

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

sage: print(CartanType(['A',1,1]).ascii_art())
O<=>O
0   1

sage: print(CartanType(['A',1,1]).ascii_art(label = lambda x: x+2))
O<=>O
2   3
dual()

Type \(A_1^1\) is self dual despite not being simply laced.

EXAMPLES:

sage: CartanType(['A',1,1]).dual()
['A', 1, 1]
dynkin_diagram()

Returns the extended Dynkin diagram for affine type A.

EXAMPLES:

sage: a = CartanType(['A',3,1]).dynkin_diagram()
sage: a
 0
 O-------+
 |       |
 |       |
 O---O---O
 1   2   3
 A3~
sage: sorted(a.edges())
[(0, 1, 1),
 (0, 3, 1),
 (1, 0, 1),
 (1, 2, 1),
 (2, 1, 1),
 (2, 3, 1),
 (3, 0, 1),
 (3, 2, 1)]

sage: a = DynkinDiagram(['A',1,1])
sage: a
O<=>O
0   1
A1~
sage: sorted(a.edges())
[(0, 1, 2), (1, 0, 2)]