Affine Weyl Groups¶
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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 affine weyl groups
TESTS:
sage: TestSuite(C).run()
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class
ElementMethods
¶ -
affine_grassmannian_to_core
()¶ Bijection between affine Grassmannian elements of type
and
-cores.
INPUT:
self
– an affine Grassmannian element of some affine Weyl group of type
Recall that an element
of an affine Weyl group is affine Grassmannian if all its all reduced words end in 0, see
is_affine_grassmannian()
.OUTPUT:
- a
-core
See also
affine_grassmannian_to_partition()
.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: Error! this only works on type 'A' affine Grassmannian elements
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affine_grassmannian_to_partition
()¶ Bijection between affine Grassmannian elements of type
and
-bounded partitions.
INPUT:
self
is affine Grassmannian element of the affine Weyl group of type(i.e. all reduced words end in 0)
OUTPUT:
-bounded partition
See also
affine_grassmannian_to_core()
.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
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is_affine_grassmannian
()¶ Tests whether
self
is affine GrassmannianAn 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 mimimal 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
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-
class
AffineWeylGroups.
ParentMethods
¶ -
affine_grassmannian_elements_of_given_length
(k)¶ Returns the affine Grassmannian elements of length
, as a list.
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]]
Todo
should return an enumerated set, with iterator, ...
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special_node
()¶ Returns the distinguished special node of the underlying Dynkin diagram
EXAMPLES:
sage: W=WeylGroup(['A',3,1]) sage: W.special_node() 0
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-
AffineWeylGroups.
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()
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AffineWeylGroups.
super_categories
()¶ EXAMPLES:
sage: AffineWeylGroups().super_categories() [Category of infinite weyl groups]