Bases: sage.structure.unique_representation.UniqueRepresentation, sage.structure.parent.Parent
An abstract class for sets of Pieri factors, used for constructing Stanley symmetric functions. The set of Pieri factors for a given type can be realized as an order ideal of the Bruhat order poset generated by a certain set of maximal elements.
See also
EXAMPLES:
sage: W = WeylGroup(['A',4])
sage: PF = W.pieri_factors()
sage: PF.an_element().reduced_word()
[4, 3, 2, 1]
sage: Waff = WeylGroup(['A',4,1])
sage: PFaff = Waff.pieri_factors()
sage: Waff.from_reduced_word(PF.an_element().reduced_word()) in PFaff
True
sage: W = WeylGroup(['B',3,1])
sage: PF = W.pieri_factors()
sage: W.from_reduced_word([2,3,2]) in PF.elements()
True
sage: PF.cardinality()
47
sage: W = WeylGroup(['C',3,1])
sage: PF = W.pieri_factors()
sage: PF.generating_series()
6*z^6 + 14*z^5 + 18*z^4 + 15*z^3 + 9*z^2 + 4*z + 1
sage: [w.reduced_word() for w in PF if w.length() == 2]
[[2, 0], [0, 1], [2, 3], [1, 2], [3, 2], [3, 1], [2, 1], [3, 0], [1, 0]]
REFERENCES:
[FoSta1994]
- Fomin, R. Stanley. Schubert polynomials and the nilCoxeter algebra. Advances in Math., 1994.
[BH1994]
- Billey, M. Haiman. Schubert polynomials for the classical groups. J. Amer. Math. Soc., 1994.
[TKLam1996] T.K. Lam. B and D analogues of stable Schubert polynomials and related insertion algorithms. PhD Thesis, MIT, 1996.
[Lam2008]
- Lam. Schubert polynomials for the affine Grassmannian. J. Amer. Math. Soc., 2008.
[LSS2009]
- Lam, A. Schilling, M. Shimozono. Schubert polynomials for the affine Grassmannian of the symplectic group. Mathematische Zeitschrift 264(4) (2010) 765-811 (arXiv:0710.2720 [math.CO])
[Pon2010]
- Pon. Types B and D affine Stanley symmetric functions, unpublished PhD Thesis, UC Davis, 2010.
Returns the function , where
is the
generator of QQ['z'].
EXAMPLES:
sage: W = WeylGroup(["A", 3, 1])
sage: weight = W.pieri_factors().default_weight()
sage: weight(1)
z
sage: weight(5)
z^5
TESTS:
sage: weight(4) in QQ['z']
True
sage: weight(0) in QQ['z']
True
sage: weight(0).parent() == QQ['z'] # todo: not implemented
True
Returns the elements of self
Those are constructed as the elements below the maximal elements of self in Bruhat order.
OUTPUT: a TransitiveIdeal object
EXAMPLES:
sage: PF = WeylGroup(['A',3]).pieri_factors()
sage: [w.reduced_word() for w in PF.elements()]
[[3, 2, 1], [2, 1], [1], [], [3, 1], [3], [3, 2], [2]]
See also
maximal_elements()
Todo
Possibly remove this method and instead have this class inherit from TransitiveIdeal.
Returns a length generating series for the elements of self
EXAMPLES:
sage: PF = WeylGroup(['C',3,1]).pieri_factors()
sage: PF.generating_series()
6*z^6 + 14*z^5 + 18*z^4 + 15*z^3 + 9*z^2 + 4*z + 1
sage: PF = WeylGroup(['B',4]).pieri_factors()
sage: PF.generating_series()
z^7 + 6*z^6 + 14*z^5 + 18*z^4 + 15*z^3 + 9*z^2 + 4*z + 1
Return the maximal length of a Pieri factor.
EXAMPLES:
In type A and A affine, this is :
sage: WeylGroup(['A',5]).pieri_factors().max_length()
5
sage: WeylGroup(['A',5,1]).pieri_factors().max_length()
5
In type B and B affine, this is :
sage: WeylGroup(['B',5,1]).pieri_factors().max_length()
9
sage: WeylGroup(['B',5]).pieri_factors().max_length()
9
In type C affine this is :
sage: WeylGroup(['C',5,1]).pieri_factors().max_length()
10
In type D affine this is :
sage: WeylGroup(['D',5,1]).pieri_factors().max_length()
8
Bases: sage.combinat.root_system.pieri_factors.PieriFactors
Base class for all parents.
Parents are the Sage/mathematical analogues of container objects in computer science.
INPUT:
If facade is specified, then Sets().Facade() is added to the categories of the parent. Furthermore, if facade is not True, the internal attribute _facade_for is set accordingly for use by Sets.Facade.ParentMethods.facade_for().
Internal invariants:
Todo
Eventually, category should be Sets by default.
TESTS:
We check that the facade option is compatible with specifying categories as a tuple:
sage: class MyClass(Parent): pass
sage: P = MyClass(facade = ZZ, category = (Monoids(), CommutativeAdditiveMonoids()))
sage: P.category()
Join of Category of monoids and Category of commutative additive monoids and Category of facade sets
This is the generic call method for all parents.
When called, it will find a map based on the Parent (or type) of x. If a coercion exists, it will always be chosen. This map will then be called (with the arguments and keywords if any).
By default this will dispatch as quickly as possible to _element_constructor_() though faster pathways are possible if so desired.
TESTS:
We check that the invariant:
self._element_init_pass_parent == guess_pass_parent(self, self._element_constructor)
is preserved (see trac ticket #5979):
sage: class MyParent(Parent):
....: def _element_constructor_(self, x):
....: print self, x
....: return sage.structure.element.Element(parent = self)
....: def _repr_(self):
....: return "my_parent"
....:
sage: my_parent = MyParent()
sage: x = my_parent("bla")
my_parent bla
sage: x.parent() # indirect doctest
my_parent
sage: x = my_parent() # shouldn't this one raise an error?
my_parent 0
sage: x = my_parent(3) # todo: not implemented why does this one fail???
my_parent 3
This function allows one to specify coercions, actions, conversions and embeddings involving this parent.
IT SHOULD ONLY BE CALLED DURING THE __INIT__ method, often at the end.
INPUT:
coerce_list – a list of coercion Morphisms to self and parents with canonical coercions to self
action_list – a list of actions on and by self
parents with conversions to self
embedding – a single Morphism from self
convert_method_name – a name to look for that other elements can implement to create elements of self (e.g. _integer_)
element_constructor – A callable object used by the __call__ method to construct new elements. Typically the element class or a bound method (defaults to self._element_constructor_).
init_no_parent – if True omit passing self in as the first argument of element_constructor for conversion. This is useful if parents are unique, or element_constructor is a bound method (this latter case can be detected automatically).
This is a multiplication method that more or less directly calls another attribute _mul_ (single underscore). This is because __mul__ can not be implemented via inheritance from the parent methods of the category, but _mul_ can be inherited. This is, e.g., used when creating twosided ideals of matrix algebras. See trac ticket #7797.
EXAMPLE:
sage: MS = MatrixSpace(QQ,2,2)
This matrix space is in fact an algebra, and in particular it is a ring, from the point of view of categories:
sage: MS.category()
Category of algebras over quotient fields
sage: MS in Rings()
True
However, its class does not inherit from the base class Ring:
sage: isinstance(MS,Ring)
False
Its _mul_ method is inherited from the category, and can be used to create a left or right ideal:
sage: MS._mul_.__module__
'sage.categories.rings'
sage: MS*MS.1 # indirect doctest
Left Ideal
(
[0 1]
[0 0]
)
of Full MatrixSpace of 2 by 2 dense matrices over Rational Field
sage: MS*[MS.1,2]
Left Ideal
(
[0 1]
[0 0],
[2 0]
[0 2]
)
of Full MatrixSpace of 2 by 2 dense matrices over Rational Field
sage: MS.1*MS
Right Ideal
(
[0 1]
[0 0]
)
of Full MatrixSpace of 2 by 2 dense matrices over Rational Field
sage: [MS.1,2]*MS
Right Ideal
(
[0 1]
[0 0],
[2 0]
[0 2]
)
of Full MatrixSpace of 2 by 2 dense matrices over Rational Field
EXAMPLES:
sage: W = WeylGroup(['C',3,1])
sage: w = W.from_reduced_word([3,2,1,0])
sage: PF = W.pieri_factors()
sage: w in PF
True
sage: w = W.from_reduced_word([1,0,1])
sage: w in PF
True
sage: w = W.from_reduced_word([1,0,1,0])
sage: w in PF
False
sage: w = W.from_reduced_word([0,1,2,3,2,1,0])
sage: w in PF
False
sage: w = W.from_reduced_word([2,0,3,2,1])
sage: w in PF
True
sage: W = WeylGroup(['B',4,1])
sage: PF = W.pieri_factors()
sage: w = W.from_reduced_word([1,2,4,3,1])
sage: w in PF
True
sage: w = W.from_reduced_word([1,2,4,3,1,0])
sage: w in PF
False
sage: w = W.from_reduced_word([2,3,4,3,2,1,0])
sage: w in PF
True
sage: W = WeylGroup(['A',4])
sage: PF = W.pieri_factors()
sage: W.from_reduced_word([4,3,1]) in PF
True
sage: W.from_reduced_word([1,2]) in PF
False
Override this method to specify coercions beyond those specified in coerce_list.
If no such coercion exists, return None or False. Otherwise, it may return either an actual Map to use for the coercion, a callable (in which case it will be wrapped in a Map), or True (in which case a generic map will be provided).
Override this method to provide additional conversions beyond those given in convert_list.
This function is called after coercions are attempted. If there is a coercion morphism in the opposite direction, one should consider adding a section method to that.
This MUST return a Map from S to self, or None. If None is returned then a generic map will be provided.
Override this method to provide an action of self on S or S on self beyond what was specified in action_list.
This must return an action which accepts an element of self and an element of S (in the order specified by self_on_left).
Returns an element of self. Want it in sufficient generality that poorly-written functions won’t work when they’re not supposed to. This is cached so doesn’t have to be super fast.
EXAMPLES:
sage: QQ._an_element_()
1/2
sage: ZZ['x,y,z']._an_element_()
x
TESTS:
Since Parent comes before the parent classes provided by categories in the hierarchy of classes, we make sure that this default implementation of _an_element_() does not override some provided by the categories. Eventually, this default implementation should be moved into the categories to avoid this workaround:
sage: S = FiniteEnumeratedSet([1,2,3])
sage: S.category()
Category of facade finite enumerated sets
sage: super(Parent, S)._an_element_
Cached version of <function _an_element_from_iterator at ...>
sage: S._an_element_()
1
sage: S = FiniteEnumeratedSet([])
sage: S._an_element_()
Traceback (most recent call last):
...
EmptySetError
Metadata about the _repr_() output.
INPUT:
Valid key arguments are:
OUTPUT:
Boolean.
EXAMPLES:
sage: ZZ._repr_option('ascii_art')
False
sage: MatrixSpace(ZZ, 2)._repr_option('element_ascii_art')
True
Initialize the category framework
Most parents initialize their category upon construction, and this is the recommended behavior. For example, this happens when the constructor calls Parent.__init__() directly or indirectly. However, some parents defer this for performance reasons. For example, sage.matrix.matrix_space.MatrixSpace does not.
EXAMPLES:
sage: P = Parent()
sage: P.category()
Category of sets
sage: class MyParent(Parent):
....: def __init__(self):
....: self._init_category_(Groups())
sage: MyParent().category()
Category of groups
Return the maximal elements of self with respect to Bruhat order.
The current implementation is via a conjectural type-free formula. Use maximal_elements_combinatorial() for proven type-specific implementations. To compare type-free and type-specific (combinatorial) implementations, use method _test_maximal_elements().
EXAMPLES:
sage: W = WeylGroup(['A',4,1])
sage: PF = W.pieri_factors()
sage: sorted([w.reduced_word() for w in PF.maximal_elements()], key=str)
[[0, 4, 3, 2], [1, 0, 4, 3], [2, 1, 0, 4], [3, 2, 1, 0], [4, 3, 2, 1]]
sage: W = WeylGroup(RootSystem(["C",3,1]).weight_space())
sage: PF = W.pieri_factors()
sage: sorted([w.reduced_word() for w in PF.maximal_elements()], key=str)
[[0, 1, 2, 3, 2, 1], [1, 0, 1, 2, 3, 2], [1, 2, 3, 2, 1, 0],
[2, 1, 0, 1, 2, 3], [2, 3, 2, 1, 0, 1], [3, 2, 1, 0, 1, 2]]
sage: W = WeylGroup(RootSystem(["B",3,1]).weight_space())
sage: PF = W.pieri_factors()
sage: sorted([w.reduced_word() for w in PF.maximal_elements()], key=str)
[[0, 2, 3, 2, 0], [1, 0, 2, 3, 2], [1, 2, 3, 2, 1],
[2, 1, 0, 2, 3], [2, 3, 2, 1, 0], [3, 2, 1, 0, 2]]
sage: W = WeylGroup(['D',4,1])
sage: PF = W.pieri_factors()
sage: sorted([w.reduced_word() for w in PF.maximal_elements()], key=str)
[[0, 2, 4, 3, 2, 0], [1, 0, 2, 4, 3, 2], [1, 2, 4, 3, 2, 1],
[2, 1, 0, 2, 4, 3], [2, 4, 3, 2, 1, 0], [3, 2, 1, 0, 2, 3],
[4, 2, 1, 0, 2, 4], [4, 3, 2, 1, 0, 2]]
Bases: sage.combinat.root_system.pieri_factors.PieriFactors
The Pieri factors of finite type A are the restriction of the Pieri factors of affine type A to finite permutations (under the canonical embedding of finite type A into the affine Weyl group), and the Pieri factors of finite type B are the restriction of the Pieri factors of affine type C. The finite type D Pieri factors are (weakly) conjectured to be the restriction of the Pieri factors of affine type D.
The current algorithm uses the fact that the maximal Pieri factors
of affine type A,B,C, or D either contain a finite Weyl group
element, or contain an affine Weyl group element whose reflection
by gets a finite Weyl group element, and that either of
these finite group elements will serve as a maximal element for
finite Pieri factors. A better algorithm is desirable.
EXAMPLES:
sage: PF = WeylGroup(['A',5]).pieri_factors()
sage: [v.reduced_word() for v in PF.maximal_elements()]
[[5, 4, 3, 2, 1]]
sage: WeylGroup(['B',4]).pieri_factors().maximal_elements()
[
[-1 0 0 0]
[ 0 1 0 0]
[ 0 0 1 0]
[ 0 0 0 1]
]
Bases: sage.combinat.root_system.pieri_factors.PieriFactors_finite_type
The set of Pieri factors for finite type A.
This is the set of elements of the Weyl group that have a reduced word that is strictly decreasing. May also be viewed as the restriction of affine type A Pieri factors to finite Weyl group elements.
Returns the maximal Pieri factors, using the type A combinatorial description
EXAMPLES:
sage: W = WeylGroup(['A',4])
sage: PF = W.pieri_factors()
sage: PF.maximal_elements_combinatorial()[0].reduced_word()
[4, 3, 2, 1]
EXAMPLES:
sage: W = WeylGroup(['A',4])
sage: PF = W.pieri_factors()
sage: PF.stanley_symm_poly_weight(W.from_reduced_word([3,1]))
0
Bases: sage.combinat.root_system.pieri_factors.PieriFactors_affine_type
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
be a Pieri factor, and
a reduced word for
. No simple
reflection appears twice in red, and the support
of
(that is the
such that
appears in
) does not depend
on the reduced word).
EXAMPLES:
sage: WeylGroup(["A", 3, 1]).pieri_factors().cardinality()
15
Returns a length generating series for the elements of self
EXAMPLES:
sage: W = WeylGroup(["A", 3, 1])
sage: W.pieri_factors().cardinality()
15
sage: W.pieri_factors().generating_series()
4*z^3 + 6*z^2 + 4*z + 1
Returns the maximal Pieri factors, using the affine type A combinatorial description
EXAMPLES:
sage: W = WeylGroup(['A',4,1])
sage: PF = W.pieri_factors()
sage: [w.reduced_word() for w in PF.maximal_elements_combinatorial()]
[[3, 2, 1, 0], [2, 1, 0, 4], [1, 0, 4, 3], [0, 4, 3, 2], [4, 3, 2, 1]]
Weight used in computing (affine) Stanley symmetric polynomials for affine type A.
EXAMPLES:
sage: W = WeylGroup(['A',5,1])
sage: PF = W.pieri_factors()
sage: PF.stanley_symm_poly_weight(W.one())
0
sage: PF.stanley_symm_poly_weight(W.from_reduced_word([5,4,2,1,0]))
0
Returns the subset of the elements of self of length length
sage: PF = WeylGroup([“A”, 3, 1]).pieri_factors(); PF Pieri factors for Weyl Group of type [‘A’, 3, 1] (as a matrix group acting on the root space) sage: PF3 = PF.subset(length = 2) sage: PF3.cardinality() 6
TESTS:
We check that there is no reference effect (there was at some point!):
sage: PF.cardinality()
15
Bases: sage.combinat.root_system.pieri_factors.PieriFactors_finite_type
The type B finite Pieri factors are realized as the set of elements that have a reduced word that is a subword of 12...(n-1)n(n-1)...21. They are the restriction of the type C affine Pieri factors to the set of finite Weyl group elements under the usual embedding.
Returns the maximal Pieri factors, using the type B combinatorial description
EXAMPLES:
sage: PF = WeylGroup(['B',4]).pieri_factors()
sage: PF.maximal_elements_combinatorial()[0].reduced_word()
[1, 2, 3, 4, 3, 2, 1]
Weight used in computing Stanley symmetric polynomials of type
. The weight for finite type B is the number of components
of the support of an element minus the number of occurrences
of
in a reduced word.
EXAMPLES:
sage: W = WeylGroup(['B',5])
sage: PF = W.pieri_factors()
sage: PF.stanley_symm_poly_weight(W.from_reduced_word([3,1,5]))
2
sage: PF.stanley_symm_poly_weight(W.from_reduced_word([3,4,5]))
0
sage: PF.stanley_symm_poly_weight(W.from_reduced_word([1,2,3,4,5,4]))
0
Bases: sage.combinat.root_system.pieri_factors.PieriFactors_affine_type
The type B affine Pieri factors are realized as the order ideal (in Bruhat order) generated by the following elements:
EXAMPLES:
sage: W = WeylGroup(['B',4,1])
sage: PF = W.pieri_factors()
sage: W.from_reduced_word([2,3,4,3,2,1,0]) in PF.maximal_elements()
True
sage: W.from_reduced_word([0,2,3,4,3,2,1]) in PF.maximal_elements()
False
sage: W.from_reduced_word([1,0,2,3,4,3,2]) in PF.maximal_elements()
True
sage: W.from_reduced_word([0,2,3,4,3,2,0]) in PF.maximal_elements()
True
sage: W.from_reduced_word([0,2,0]) in PF
True
Returns the maximal Pieri factors, using the affine type B combinatorial description
EXAMPLES:
sage: W = WeylGroup(['B',4,1])
sage: [u.reduced_word() for u in W.pieri_factors().maximal_elements_combinatorial()]
[[1, 0, 2, 3, 4, 3, 2], [2, 1, 0, 2, 3, 4, 3], [3, 2, 1, 0, 2, 3, 4], [4, 3, 2, 1, 0, 2, 3], [3, 4, 3, 2, 1, 0, 2], [2, 3, 4, 3, 2, 1, 0], [1, 2, 3, 4, 3, 2, 1], [0, 2, 3, 4, 3, 2, 0]]
Returns the weight of a Pieri factor to be used in the definition of Stanley symmetric functions. For type B, this weight involves the number of components of the complement of the support of an element, where we consider 0 and 1 to be one node – if 1 is in the support, then we pretend 0 in the support, and vice versa. We also consider 0 and 1 to be one node for the purpose of counting components of the complement (as if the Dynkin diagram were that of type C). Let n be the rank of the affine Weyl group in question (if type [‘B’,k,1] then we have n = k+1). Let chi(v.length() < n-1) be the indicator function that is 1 if the length of v is smaller than n-1, and 0 if the length of v is greater than or equal to n-1. If we say c’(v) = the number of components of the complement of the support of v, then the type B weight is given by weight = c’(v) - chi(v.length() < n-1).
EXAMPLES:
sage: W = WeylGroup(['B',5,1])
sage: PF = W.pieri_factors()
sage: PF.stanley_symm_poly_weight(W.from_reduced_word([0,3]))
1
sage: PF.stanley_symm_poly_weight(W.from_reduced_word([0,1,3]))
1
sage: PF.stanley_symm_poly_weight(W.from_reduced_word([2,3]))
1
sage: PF.stanley_symm_poly_weight(W.from_reduced_word([2,3,4,5]))
0
sage: PF.stanley_symm_poly_weight(W.from_reduced_word([0,5]))
0
sage: PF.stanley_symm_poly_weight(W.from_reduced_word([2,4,5,4,3,0]))
-1
sage: PF.stanley_symm_poly_weight(W.from_reduced_word([4,5,4,3,0]))
0
Bases: sage.combinat.root_system.pieri_factors.PieriFactors_affine_type
The type C affine Pieri factors are realized as the order ideal (in Bruhat order) generated by cyclic rotations of the element with unique reduced word 123...(n-1)n(n-1)...3210.
EXAMPLES:
sage: W = WeylGroup(['C',3,1])
sage: PF = W.pieri_factors()
sage: sorted([u.reduced_word() for u in PF.maximal_elements()], key=str)
[[0, 1, 2, 3, 2, 1], [1, 0, 1, 2, 3, 2], [1, 2, 3, 2, 1, 0],
[2, 1, 0, 1, 2, 3], [2, 3, 2, 1, 0, 1], [3, 2, 1, 0, 1, 2]]
Returns the maximal Pieri factors, using the affine type C combinatorial description
EXAMPLES:
sage: PF = WeylGroup(['C',3,1]).pieri_factors()
sage: [w.reduced_word() for w in PF.maximal_elements_combinatorial()]
[[0, 1, 2, 3, 2, 1], [1, 0, 1, 2, 3, 2], [2, 1, 0, 1, 2, 3], [3, 2, 1, 0, 1, 2], [2, 3, 2, 1, 0, 1], [1, 2, 3, 2, 1, 0]]
Returns the weight of a Pieri factor to be used in the definition of Stanley symmetric functions. For type C, this weight is the number of connected components of the support (the indices appearing in a reduced word) of an element.
EXAMPLES:
sage: W = WeylGroup(['C',5,1])
sage: PF = W.pieri_factors()
sage: PF.stanley_symm_poly_weight(W.from_reduced_word([1,3]))
2
sage: PF.stanley_symm_poly_weight(W.from_reduced_word([1,3,2,0]))
1
sage: PF.stanley_symm_poly_weight(W.from_reduced_word([5,3,0]))
3
sage: PF.stanley_symm_poly_weight(W.one())
0
Bases: sage.combinat.root_system.pieri_factors.PieriFactors_affine_type
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]]
Returns the maximal Pieri factors, using the affine type D combinatorial description
EXAMPLES:
sage: W = WeylGroup(['D',5,1])
sage: PF = W.pieri_factors()
sage: set(PF.maximal_elements_combinatorial()) == set(PF.maximal_elements())
True
Returns the weight of , to be used in the definition of
Stanley symmetric functions. For type D, this weight involves
the number of components of the complement of the support of
an element, where we consider 0 and 1 to be one node – if 1
is in the support, then we pretend 0 in the support, and vice
versa. Similarly with
and
. We also consider 0 and
1, n-1 and n to be one node for the purpose of counting
components of the complement (as if the Dynkin diagram were
that of type C).
Type D Stanley symmetric polynomial weights are still conjectural. The given weight comes from conditions on elements of the affine Fomin-Stanley subalgebra, but work is needed to show this weight is correct for affine Stanley symmetric functions – see [LSS2009, Pon2010] for details.
EXAMPLES:
sage: W = WeylGroup(['D', 5, 1])
sage: PF = W.pieri_factors()
sage: PF.stanley_symm_poly_weight(W.from_reduced_word([5,2,1]))
0
sage: PF.stanley_symm_poly_weight(W.from_reduced_word([5,2,1,0]))
0
sage: PF.stanley_symm_poly_weight(W.from_reduced_word([5,2]))
1
sage: PF.stanley_symm_poly_weight(W.from_reduced_word([]))
0
sage: W = WeylGroup(['D',7,1])
sage: PF = W.pieri_factors()
sage: PF.stanley_symm_poly_weight(W.from_reduced_word([2,4,6]))
2