Affine Weyl groups¶

class
sage.categories.affine_weyl_groups.
AffineWeylGroups
(s=None)¶ Bases:
sage.categories.category_singleton.Category_singleton
The category of affine Weyl groups
Todo
add a description of this category
See also
 Wikipedia article Affine_weyl_group
WeylGroups
,WeylGroup
EXAMPLES:
sage: C = AffineWeylGroups(); C Category of affine weyl groups sage: C.super_categories() [Category of infinite weyl groups] sage: C.example() NotImplemented sage: W = WeylGroup(["A",4,1]); W Weyl Group of type ['A', 4, 1] (as a matrix group acting on the root space) sage: W.category() Category of irreducible affine weyl groups

class
ElementMethods
¶ 
affine_grassmannian_to_core
()¶ Bijection between affine Grassmannian elements of type \(A_k^{(1)}\) and \((k+1)\)cores.
INPUT:
self
– an affine Grassmannian element of some affine Weyl group of type \(A_k^{(1)}\)
Recall that an element \(w\) of an affine Weyl group is affine Grassmannian if all its all reduced words end in 0, see
is_affine_grassmannian()
.OUTPUT:
 a \((k+1)\)core
See also
EXAMPLES:
sage: W = WeylGroup(['A',2,1]) sage: w = W.from_reduced_word([0,2,1,0]) sage: la = w.affine_grassmannian_to_core(); la [4, 2] sage: type(la) <class 'sage.combinat.core.Cores_length_with_category.element_class'> sage: la.to_grassmannian() == w True sage: w = W.from_reduced_word([0,2,1]) sage: w.affine_grassmannian_to_core() Traceback (most recent call last): ... ValueError: this only works on type 'A' affine Grassmannian elements

affine_grassmannian_to_partition
()¶ Bijection between affine Grassmannian elements of type \(A_k^{(1)}\) and \(k\)bounded partitions.
INPUT:
self
is affine Grassmannian element of the affine Weyl group of type \(A_k^{(1)}\) (i.e. all reduced words end in 0)
OUTPUT:
 \(k\)bounded partition
See also
EXAMPLES:
sage: k = 2 sage: W = WeylGroup(['A',k,1]) sage: w = W.from_reduced_word([0,2,1,0]) sage: la = w.affine_grassmannian_to_partition(); la [2, 2] sage: la.from_kbounded_to_grassmannian(k) == w True

is_affine_grassmannian
()¶ Test whether
self
is affine Grassmannian.An element of an affine Weyl group is affine Grassmannian if any of the following equivalent properties holds:
 all reduced words for
self
end with 0. self
is the identity, or 0 is its single right descent.self
is a minimal coset representative for W / cl W.
EXAMPLES:
sage: W = WeylGroup(['A',3,1]) sage: w = W.from_reduced_word([2,1,0]) sage: w.is_affine_grassmannian() True sage: w = W.from_reduced_word([2,0]) sage: w.is_affine_grassmannian() False sage: W.one().is_affine_grassmannian() True
 all reduced words for


class
ParentMethods
¶ 
affine_grassmannian_elements_of_given_length
(k)¶ Return the affine Grassmannian elements of length \(k\).
This is returned as a finite enumerated set.
EXAMPLES:
sage: W = WeylGroup(['A',3,1]) sage: [x.reduced_word() for x in W.affine_grassmannian_elements_of_given_length(3)] [[2, 1, 0], [3, 1, 0], [2, 3, 0]]

special_node
()¶ Return the distinguished special node of the underlying Dynkin diagram.
EXAMPLES:
sage: W = WeylGroup(['A',3,1]) sage: W.special_node() 0


additional_structure
()¶ Return
None
.Indeed, the category of affine Weyl groups defines no additional structure: affine Weyl groups are a special class of Weyl groups.
See also
Todo
Should this category be a
CategoryWithAxiom
?EXAMPLES:
sage: AffineWeylGroups().additional_structure()

super_categories
()¶ EXAMPLES:
sage: AffineWeylGroups().super_categories() [Category of infinite weyl groups]