Pieri Factors

class sage.combinat.root_system.pieri_factors.PieriFactors

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.

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]S. Fomin, R. Stanley. Schubert polynomials and the nilCoxeter algebra. Advances in Math., 1994.
[BH1994]S. 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]T. Lam. Schubert polynomials for the affine Grassmannian. J. Amer. Math. Soc., 2008.
[LSS2009]T. 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]S. Pon. Types B and D affine Stanley symmetric functions, unpublished PhD Thesis, UC Davis, 2010.
default_weight()

Returns the function \(i\mapsto z^i\), where \(z\) 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
elements()

Returns the elements of self

Those are constructed as the elements below the maximal elements of self in Bruhat order.

OUTPUT: a RecursivelyEnumeratedSet_generic 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 RecursivelyEnumeratedSet_generic.

generating_series(weight=None)

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
max_length()

Return the maximal length of a Pieri factor.

EXAMPLES:

In type A and A affine, this is \(n\):

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 \(2n-1\):

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 \(2n\):

sage: WeylGroup(['C',5,1]).pieri_factors().max_length()
10

In type D affine this is \(2n-2\):

sage: WeylGroup(['D',5,1]).pieri_factors().max_length()
8
class sage.combinat.root_system.pieri_factors.PieriFactors_affine_type

Bases: sage.combinat.root_system.pieri_factors.PieriFactors

maximal_elements()

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]]
class sage.combinat.root_system.pieri_factors.PieriFactors_finite_type

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.

maximal_elements()

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 \(s_0\) 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]
]
class sage.combinat.root_system.pieri_factors.PieriFactors_type_A(W)

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.

maximal_elements_combinatorial()

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]
stanley_symm_poly_weight(w)

EXAMPLES:

sage: W = WeylGroup(['A',4])
sage: PF = W.pieri_factors()
sage: PF.stanley_symm_poly_weight(W.from_reduced_word([3,1]))
0
class sage.combinat.root_system.pieri_factors.PieriFactors_type_A_affine(W, min_length, max_length, min_support, max_support)

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 \(f\) 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).

cardinality()

EXAMPLES:

sage: WeylGroup(["A", 3, 1]).pieri_factors().cardinality()
15
generating_series(weight=None)

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
maximal_elements_combinatorial()

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]]
stanley_symm_poly_weight(w)

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
subset(length)
INPUT:
  • length – a non-negative integer

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
class sage.combinat.root_system.pieri_factors.PieriFactors_type_B(W)

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.

maximal_elements_combinatorial()

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]
stanley_symm_poly_weight(w)

Weight used in computing Stanley symmetric polynomials of type \(B\). The weight for finite type B is the number of components of the support of an element minus the number of occurrences of \(n\) 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
class sage.combinat.root_system.pieri_factors.PieriFactors_type_B_affine(W)

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:

  • cyclic rotations of the element with reduced word 234...(n-1)n(n-1)...3210, except for 123...n...320 and 023...n...321.
  • 123...(n-1)n(n-1)...321
  • 023...(n-1)n(n-1)...320

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
maximal_elements_combinatorial()

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]]
stanley_symm_poly_weight(w)

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
class sage.combinat.root_system.pieri_factors.PieriFactors_type_C_affine(W)

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]]
maximal_elements_combinatorial()

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]]
stanley_symm_poly_weight(w)

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
class sage.combinat.root_system.pieri_factors.PieriFactors_type_D_affine(W)

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]]
maximal_elements_combinatorial()

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
stanley_symm_poly_weight(w)
INPUT:
  • w – a pieri factor for this type

Returns the weight of \(w\), 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 \(n-1\) and \(n\). 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