Weyl Groups

class sage.categories.weyl_groups.WeylGroups

Bases: Category_singleton

The category of Weyl groups.

See the Wikipedia page of Weyl Groups.

EXAMPLES:

Sage

sage: WeylGroups()
Category of Weyl groups
sage: WeylGroups().super_categories()
[Category of Coxeter groups]

Python

>>> from sage.all import *
>>> WeylGroups()
Category of Weyl groups
>>> WeylGroups().super_categories()
[Category of Coxeter groups]

Sage Live

WeylGroups()
WeylGroups().super_categories()

Here are some examples:

Sage

sage: WeylGroups().example()            # todo: not implemented
sage: FiniteWeylGroups().example()
The symmetric group on {0, ..., 3}
sage: AffineWeylGroups().example()      # todo: not implemented
sage: WeylGroup(["B", 3])
Weyl Group of type ['B', 3] (as a matrix group acting on the ambient space)

Python

>>> from sage.all import *
>>> WeylGroups().example()            # todo: not implemented
>>> FiniteWeylGroups().example()
The symmetric group on {0, ..., 3}
>>> AffineWeylGroups().example()      # todo: not implemented
>>> WeylGroup(["B", Integer(3)])
Weyl Group of type ['B', 3] (as a matrix group acting on the ambient space)

Sage Live

WeylGroups().example()            # todo: not implemented
FiniteWeylGroups().example()
AffineWeylGroups().example()      # todo: not implemented
WeylGroup(["B", 3])

This one will eventually be also in this category:

Sage

sage: SymmetricGroup(4)
Symmetric group of order 4! as a permutation group

Python

>>> from sage.all import *
>>> SymmetricGroup(Integer(4))
Symmetric group of order 4! as a permutation group

Sage Live

SymmetricGroup(4)

class ElementMethods

Bases: object

bruhat_lower_covers_coroots()

Return all 2-tuples (v, \(\alpha\)) where v is covered by self and \(\alpha\) is the positive coroot such that self = v \(s_\alpha\) where \(s_\alpha\) is the reflection orthogonal to \(\alpha\).

ALGORITHM:

See bruhat_lower_covers() and bruhat_lower_covers_reflections() for Coxeter groups.

EXAMPLES:

Sage

sage: W = WeylGroup(['A',3], prefix='s')
sage: w = W.from_reduced_word([3,1,2,1])
sage: w.bruhat_lower_covers_coroots()
[(s1*s2*s1, alphacheck[1] + alphacheck[2] + alphacheck[3]),
 (s3*s2*s1, alphacheck[2]), (s3*s1*s2, alphacheck[1])]

Python

>>> from sage.all import *
>>> W = WeylGroup(['A',Integer(3)], prefix='s')
>>> w = W.from_reduced_word([Integer(3),Integer(1),Integer(2),Integer(1)])
>>> w.bruhat_lower_covers_coroots()
[(s1*s2*s1, alphacheck[1] + alphacheck[2] + alphacheck[3]),
 (s3*s2*s1, alphacheck[2]), (s3*s1*s2, alphacheck[1])]

Sage Live

W = WeylGroup(['A',3], prefix='s')
w = W.from_reduced_word([3,1,2,1])
w.bruhat_lower_covers_coroots()

bruhat_upper_covers_coroots()

Return all 2-tuples (v, \(\alpha\)) where v is covers self and \(\alpha\) is the positive coroot such that self = v \(s_\alpha\) where \(s_\alpha\) is the reflection orthogonal to \(\alpha\).

ALGORITHM:

See bruhat_upper_covers() and bruhat_upper_covers_reflections() for Coxeter groups.

EXAMPLES:

Sage

sage: W = WeylGroup(['A',4], prefix='s')
sage: w = W.from_reduced_word([3,1,2,1])
sage: w.bruhat_upper_covers_coroots()
[(s1*s2*s3*s2*s1, alphacheck[3]),
 (s2*s3*s1*s2*s1, alphacheck[2] + alphacheck[3]),
 (s3*s4*s1*s2*s1, alphacheck[4]),
 (s4*s3*s1*s2*s1, alphacheck[1] + alphacheck[2] + alphacheck[3] + alphacheck[4])]

Python

>>> from sage.all import *
>>> W = WeylGroup(['A',Integer(4)], prefix='s')
>>> w = W.from_reduced_word([Integer(3),Integer(1),Integer(2),Integer(1)])
>>> w.bruhat_upper_covers_coroots()
[(s1*s2*s3*s2*s1, alphacheck[3]),
 (s2*s3*s1*s2*s1, alphacheck[2] + alphacheck[3]),
 (s3*s4*s1*s2*s1, alphacheck[4]),
 (s4*s3*s1*s2*s1, alphacheck[1] + alphacheck[2] + alphacheck[3] + alphacheck[4])]

Sage Live

W = WeylGroup(['A',4], prefix='s')
w = W.from_reduced_word([3,1,2,1])
w.bruhat_upper_covers_coroots()

inversion_arrangement(side='right')

Return the inversion hyperplane arrangement of self.

INPUT:

• side – 'right' (default) or 'left'

OUTPUT:

A (central) hyperplane arrangement whose hyperplanes correspond to the inversions of self given as roots.

The side parameter determines on which side to compute the inversions.

EXAMPLES:

Sage

sage: W = WeylGroup(['A',3])
sage: w = W.from_reduced_word([1, 2, 3, 1, 2])
sage: A = w.inversion_arrangement(); A
Arrangement of 5 hyperplanes of dimension 3 and rank 3
sage: A.hyperplanes()
(Hyperplane 0*a1 + 0*a2 + a3 + 0,
 Hyperplane 0*a1 + a2 + 0*a3 + 0,
 Hyperplane 0*a1 + a2 + a3 + 0,
 Hyperplane a1 + a2 + 0*a3 + 0,
 Hyperplane a1 + a2 + a3 + 0)

Python

>>> from sage.all import *
>>> W = WeylGroup(['A',Integer(3)])
>>> w = W.from_reduced_word([Integer(1), Integer(2), Integer(3), Integer(1), Integer(2)])
>>> A = w.inversion_arrangement(); A
Arrangement of 5 hyperplanes of dimension 3 and rank 3
>>> A.hyperplanes()
(Hyperplane 0*a1 + 0*a2 + a3 + 0,
 Hyperplane 0*a1 + a2 + 0*a3 + 0,
 Hyperplane 0*a1 + a2 + a3 + 0,
 Hyperplane a1 + a2 + 0*a3 + 0,
 Hyperplane a1 + a2 + a3 + 0)

Sage Live

W = WeylGroup(['A',3])
w = W.from_reduced_word([1, 2, 3, 1, 2])
A = w.inversion_arrangement(); A
A.hyperplanes()

The identity element gives the empty arrangement:

Sage

sage: W = WeylGroup(['A',3])
sage: W.one().inversion_arrangement()
Empty hyperplane arrangement of dimension 3

Python

>>> from sage.all import *
>>> W = WeylGroup(['A',Integer(3)])
>>> W.one().inversion_arrangement()
Empty hyperplane arrangement of dimension 3

Sage Live

W = WeylGroup(['A',3])
W.one().inversion_arrangement()

inversions(side='right', inversion_type='reflections')

Return the set of inversions of self.

INPUT:

• side – ‘right’ (default) or ‘left’

• inversion_type – ‘reflections’ (default), ‘roots’, or ‘coroots’

OUTPUT:

For reflections, the set of reflections r in the Weyl group such that self r < self. For (co)roots, the set of positive (co)roots that are sent by self to negative (co)roots; their associated reflections are described above.

If side is ‘left’, the inverse Weyl group element is used.

EXAMPLES:

Sage

sage: W = WeylGroup(['C',2], prefix='s')
sage: w = W.from_reduced_word([1,2])
sage: w.inversions()
[s2, s2*s1*s2]
sage: w.inversions(inversion_type = 'reflections')
[s2, s2*s1*s2]
sage: w.inversions(inversion_type = 'roots')
[alpha[2], alpha[1] + alpha[2]]
sage: w.inversions(inversion_type = 'coroots')
[alphacheck[2], alphacheck[1] + 2*alphacheck[2]]
sage: w.inversions(side = 'left')
[s1, s1*s2*s1]
sage: w.inversions(side = 'left', inversion_type = 'roots')
[alpha[1], 2*alpha[1] + alpha[2]]
sage: w.inversions(side = 'left', inversion_type = 'coroots')
[alphacheck[1], alphacheck[1] + alphacheck[2]]

Python

>>> from sage.all import *
>>> W = WeylGroup(['C',Integer(2)], prefix='s')
>>> w = W.from_reduced_word([Integer(1),Integer(2)])
>>> w.inversions()
[s2, s2*s1*s2]
>>> w.inversions(inversion_type = 'reflections')
[s2, s2*s1*s2]
>>> w.inversions(inversion_type = 'roots')
[alpha[2], alpha[1] + alpha[2]]
>>> w.inversions(inversion_type = 'coroots')
[alphacheck[2], alphacheck[1] + 2*alphacheck[2]]
>>> w.inversions(side = 'left')
[s1, s1*s2*s1]
>>> w.inversions(side = 'left', inversion_type = 'roots')
[alpha[1], 2*alpha[1] + alpha[2]]
>>> w.inversions(side = 'left', inversion_type = 'coroots')
[alphacheck[1], alphacheck[1] + alphacheck[2]]

Sage Live

W = WeylGroup(['C',2], prefix='s')
w = W.from_reduced_word([1,2])
w.inversions()
w.inversions(inversion_type = 'reflections')
w.inversions(inversion_type = 'roots')
w.inversions(inversion_type = 'coroots')
w.inversions(side = 'left')
w.inversions(side = 'left', inversion_type = 'roots')
w.inversions(side = 'left', inversion_type = 'coroots')

is_pieri_factor()

Return whether self is a Pieri factor, as used for computing Stanley symmetric functions.

See also

• stanley_symmetric_function()

• WeylGroups.ParentMethods.pieri_factors()

EXAMPLES:

Sage

sage: W = WeylGroup(['A',5,1])
sage: W.from_reduced_word([3,2,5]).is_pieri_factor()
True
sage: W.from_reduced_word([3,2,4,5]).is_pieri_factor()
False

sage: W = WeylGroup(['C',4,1])
sage: W.from_reduced_word([0,2,1]).is_pieri_factor()
True
sage: W.from_reduced_word([0,2,1,0]).is_pieri_factor()
False

sage: W = WeylGroup(['B',3])
sage: W.from_reduced_word([3,2,3]).is_pieri_factor()
False
sage: W.from_reduced_word([2,1,2]).is_pieri_factor()
True

Python

>>> from sage.all import *
>>> W = WeylGroup(['A',Integer(5),Integer(1)])
>>> W.from_reduced_word([Integer(3),Integer(2),Integer(5)]).is_pieri_factor()
True
>>> W.from_reduced_word([Integer(3),Integer(2),Integer(4),Integer(5)]).is_pieri_factor()
False

>>> W = WeylGroup(['C',Integer(4),Integer(1)])
>>> W.from_reduced_word([Integer(0),Integer(2),Integer(1)]).is_pieri_factor()
True
>>> W.from_reduced_word([Integer(0),Integer(2),Integer(1),Integer(0)]).is_pieri_factor()
False

>>> W = WeylGroup(['B',Integer(3)])
>>> W.from_reduced_word([Integer(3),Integer(2),Integer(3)]).is_pieri_factor()
False
>>> W.from_reduced_word([Integer(2),Integer(1),Integer(2)]).is_pieri_factor()
True

Sage Live

W = WeylGroup(['A',5,1])
W.from_reduced_word([3,2,5]).is_pieri_factor()
W.from_reduced_word([3,2,4,5]).is_pieri_factor()
W = WeylGroup(['C',4,1])
W.from_reduced_word([0,2,1]).is_pieri_factor()
W.from_reduced_word([0,2,1,0]).is_pieri_factor()
W = WeylGroup(['B',3])
W.from_reduced_word([3,2,3]).is_pieri_factor()
W.from_reduced_word([2,1,2]).is_pieri_factor()

left_pieri_factorizations(max_length=None)

Return all factorizations of self as \(uv\), where \(u\) is a Pieri factor and \(v\) is an element of the Weyl group.

See also

• WeylGroups.ParentMethods.pieri_factors()

• sage.combinat.root_system.pieri_factors

EXAMPLES:

If we take \(w = w_0\) the maximal element of a strict parabolic subgroup of type \(A_{n_1} \times \cdots \times A_{n_k}\), then the Pieri factorizations are in correspondence with all Pieri factors, and there are \(\prod 2^{n_i}\) of them:

Sage

sage: W = WeylGroup(['A', 4, 1])
sage: W.from_reduced_word([]).left_pieri_factorizations().cardinality()
1
sage: W.from_reduced_word([1]).left_pieri_factorizations().cardinality()
2
sage: W.from_reduced_word([1,2,1]).left_pieri_factorizations().cardinality()
4
sage: W.from_reduced_word([1,2,3,1,2,1]).left_pieri_factorizations().cardinality()
8

sage: W.from_reduced_word([1,3]).left_pieri_factorizations().cardinality()
4
sage: W.from_reduced_word([1,3,4,3]).left_pieri_factorizations().cardinality()
8

sage: W.from_reduced_word([2,1]).left_pieri_factorizations().cardinality()
3
sage: W.from_reduced_word([1,2]).left_pieri_factorizations().cardinality()
2
sage: [W.from_reduced_word([1,2]).left_pieri_factorizations(max_length=i).cardinality()
....:  for i in [-1, 0, 1, 2]]
[0, 1, 2, 2]

sage: W = WeylGroup(['C',4,1])
sage: w = W.from_reduced_word([0,3,2,1,0])
sage: w.left_pieri_factorizations().cardinality()
7
sage: [(u.reduced_word(),v.reduced_word())
....:  for (u,v) in w.left_pieri_factorizations()]
[([], [3, 2, 0, 1, 0]),
([0], [3, 2, 1, 0]),
([3], [2, 0, 1, 0]),
([3, 0], [2, 1, 0]),
([3, 2], [0, 1, 0]),
([3, 2, 0], [1, 0]),
([3, 2, 0, 1], [0])]

sage: W = WeylGroup(['B',4,1])
sage: W.from_reduced_word([0,2,1,0]).left_pieri_factorizations().cardinality()
6

Python

>>> from sage.all import *
>>> W = WeylGroup(['A', Integer(4), Integer(1)])
>>> W.from_reduced_word([]).left_pieri_factorizations().cardinality()
1
>>> W.from_reduced_word([Integer(1)]).left_pieri_factorizations().cardinality()
2
>>> W.from_reduced_word([Integer(1),Integer(2),Integer(1)]).left_pieri_factorizations().cardinality()
4
>>> W.from_reduced_word([Integer(1),Integer(2),Integer(3),Integer(1),Integer(2),Integer(1)]).left_pieri_factorizations().cardinality()
8

>>> W.from_reduced_word([Integer(1),Integer(3)]).left_pieri_factorizations().cardinality()
4
>>> W.from_reduced_word([Integer(1),Integer(3),Integer(4),Integer(3)]).left_pieri_factorizations().cardinality()
8

>>> W.from_reduced_word([Integer(2),Integer(1)]).left_pieri_factorizations().cardinality()
3
>>> W.from_reduced_word([Integer(1),Integer(2)]).left_pieri_factorizations().cardinality()
2
>>> [W.from_reduced_word([Integer(1),Integer(2)]).left_pieri_factorizations(max_length=i).cardinality()
...  for i in [-Integer(1), Integer(0), Integer(1), Integer(2)]]
[0, 1, 2, 2]

>>> W = WeylGroup(['C',Integer(4),Integer(1)])
>>> w = W.from_reduced_word([Integer(0),Integer(3),Integer(2),Integer(1),Integer(0)])
>>> w.left_pieri_factorizations().cardinality()
7
>>> [(u.reduced_word(),v.reduced_word())
...  for (u,v) in w.left_pieri_factorizations()]
[([], [3, 2, 0, 1, 0]),
([0], [3, 2, 1, 0]),
([3], [2, 0, 1, 0]),
([3, 0], [2, 1, 0]),
([3, 2], [0, 1, 0]),
([3, 2, 0], [1, 0]),
([3, 2, 0, 1], [0])]

>>> W = WeylGroup(['B',Integer(4),Integer(1)])
>>> W.from_reduced_word([Integer(0),Integer(2),Integer(1),Integer(0)]).left_pieri_factorizations().cardinality()
6

Sage Live

W = WeylGroup(['A', 4, 1])
W.from_reduced_word([]).left_pieri_factorizations().cardinality()
W.from_reduced_word([1]).left_pieri_factorizations().cardinality()
W.from_reduced_word([1,2,1]).left_pieri_factorizations().cardinality()
W.from_reduced_word([1,2,3,1,2,1]).left_pieri_factorizations().cardinality()
W.from_reduced_word([1,3]).left_pieri_factorizations().cardinality()
W.from_reduced_word([1,3,4,3]).left_pieri_factorizations().cardinality()
W.from_reduced_word([2,1]).left_pieri_factorizations().cardinality()
W.from_reduced_word([1,2]).left_pieri_factorizations().cardinality()
[W.from_reduced_word([1,2]).left_pieri_factorizations(max_length=i).cardinality()
 for i in [-1, 0, 1, 2]]
W = WeylGroup(['C',4,1])
w = W.from_reduced_word([0,3,2,1,0])
w.left_pieri_factorizations().cardinality()
[(u.reduced_word(),v.reduced_word())
 for (u,v) in w.left_pieri_factorizations()]
W = WeylGroup(['B',4,1])
W.from_reduced_word([0,2,1,0]).left_pieri_factorizations().cardinality()

quantum_bruhat_successors(index_set=None, roots=False, quantum_only=False)

Return the successors of self in the quantum Bruhat graph on the parabolic quotient of the Weyl group determined by the subset of Dynkin nodes index_set.

INPUT:

• self – a Weyl group element, which is assumed to be of minimum length in its coset with respect to the parabolic subgroup

• index_set – (default: None) indicates the set of simple reflections used to generate the parabolic subgroup; the default value indicates that the subgroup is the identity

• roots – boolean (default: False); if True, returns the list of 2-tuples (w, \(\alpha\)) where w is a successor and \(\alpha\) is the positive root associated with the successor relation

• quantum_only – boolean (default: False); if True, returns only the quantum successors

EXAMPLES:

Sage

sage: W = WeylGroup(['A',3], prefix='s')
sage: w = W.from_reduced_word([3,1,2])
sage: w.quantum_bruhat_successors([1], roots = True)
[(s3, alpha[2]), (s1*s2*s3*s2, alpha[3]),
 (s2*s3*s1*s2, alpha[1] + alpha[2] + alpha[3])]
sage: w.quantum_bruhat_successors([1,3])
[1, s2*s3*s1*s2]
sage: w.quantum_bruhat_successors(roots = True)
[(s3*s1*s2*s1, alpha[1]),
 (s3*s1, alpha[2]),
 (s1*s2*s3*s2, alpha[3]),
 (s2*s3*s1*s2, alpha[1] + alpha[2] + alpha[3])]
sage: w.quantum_bruhat_successors()
[s3*s1*s2*s1, s3*s1, s1*s2*s3*s2, s2*s3*s1*s2]
sage: w.quantum_bruhat_successors(quantum_only = True)
[s3*s1]
sage: w = W.from_reduced_word([2,3])
sage: w.quantum_bruhat_successors([1,3])
Traceback (most recent call last):
...
ValueError: s2*s3 is not of minimum length in its coset
of the parabolic subgroup generated by the reflections (1, 3)

Python

>>> from sage.all import *
>>> W = WeylGroup(['A',Integer(3)], prefix='s')
>>> w = W.from_reduced_word([Integer(3),Integer(1),Integer(2)])
>>> w.quantum_bruhat_successors([Integer(1)], roots = True)
[(s3, alpha[2]), (s1*s2*s3*s2, alpha[3]),
 (s2*s3*s1*s2, alpha[1] + alpha[2] + alpha[3])]
>>> w.quantum_bruhat_successors([Integer(1),Integer(3)])
[1, s2*s3*s1*s2]
>>> w.quantum_bruhat_successors(roots = True)
[(s3*s1*s2*s1, alpha[1]),
 (s3*s1, alpha[2]),
 (s1*s2*s3*s2, alpha[3]),
 (s2*s3*s1*s2, alpha[1] + alpha[2] + alpha[3])]
>>> w.quantum_bruhat_successors()
[s3*s1*s2*s1, s3*s1, s1*s2*s3*s2, s2*s3*s1*s2]
>>> w.quantum_bruhat_successors(quantum_only = True)
[s3*s1]
>>> w = W.from_reduced_word([Integer(2),Integer(3)])
>>> w.quantum_bruhat_successors([Integer(1),Integer(3)])
Traceback (most recent call last):
...
ValueError: s2*s3 is not of minimum length in its coset
of the parabolic subgroup generated by the reflections (1, 3)

Sage Live

W = WeylGroup(['A',3], prefix='s')
w = W.from_reduced_word([3,1,2])
w.quantum_bruhat_successors([1], roots = True)
w.quantum_bruhat_successors([1,3])
w.quantum_bruhat_successors(roots = True)
w.quantum_bruhat_successors()
w.quantum_bruhat_successors(quantum_only = True)
w = W.from_reduced_word([2,3])
w.quantum_bruhat_successors([1,3])

reflection_to_coroot()

Return the coroot associated with the reflection self.

EXAMPLES:

Sage

sage: W = WeylGroup(['C',2], prefix='s')
sage: W.from_reduced_word([1,2,1]).reflection_to_coroot()
alphacheck[1] + alphacheck[2]
sage: W.from_reduced_word([1,2]).reflection_to_coroot()
Traceback (most recent call last):
...
ValueError: s1*s2 is not a reflection
sage: W.long_element().reflection_to_coroot()
Traceback (most recent call last):
...
ValueError: s2*s1*s2*s1 is not a reflection

Python

>>> from sage.all import *
>>> W = WeylGroup(['C',Integer(2)], prefix='s')
>>> W.from_reduced_word([Integer(1),Integer(2),Integer(1)]).reflection_to_coroot()
alphacheck[1] + alphacheck[2]
>>> W.from_reduced_word([Integer(1),Integer(2)]).reflection_to_coroot()
Traceback (most recent call last):
...
ValueError: s1*s2 is not a reflection
>>> W.long_element().reflection_to_coroot()
Traceback (most recent call last):
...
ValueError: s2*s1*s2*s1 is not a reflection

Sage Live

W = WeylGroup(['C',2], prefix='s')
W.from_reduced_word([1,2,1]).reflection_to_coroot()
W.from_reduced_word([1,2]).reflection_to_coroot()
W.long_element().reflection_to_coroot()

reflection_to_root()

Return the root associated with the reflection self.

EXAMPLES:

Sage

sage: W = WeylGroup(['C',2], prefix='s')
sage: W.from_reduced_word([1,2,1]).reflection_to_root()
2*alpha[1] + alpha[2]
sage: W.from_reduced_word([1,2]).reflection_to_root()
Traceback (most recent call last):
...
ValueError: s1*s2 is not a reflection
sage: W.long_element().reflection_to_root()
Traceback (most recent call last):
...
ValueError: s2*s1*s2*s1 is not a reflection

Python

>>> from sage.all import *
>>> W = WeylGroup(['C',Integer(2)], prefix='s')
>>> W.from_reduced_word([Integer(1),Integer(2),Integer(1)]).reflection_to_root()
2*alpha[1] + alpha[2]
>>> W.from_reduced_word([Integer(1),Integer(2)]).reflection_to_root()
Traceback (most recent call last):
...
ValueError: s1*s2 is not a reflection
>>> W.long_element().reflection_to_root()
Traceback (most recent call last):
...
ValueError: s2*s1*s2*s1 is not a reflection

Sage Live

W = WeylGroup(['C',2], prefix='s')
W.from_reduced_word([1,2,1]).reflection_to_root()
W.from_reduced_word([1,2]).reflection_to_root()
W.long_element().reflection_to_root()

stanley_symmetric_function()

Return the affine Stanley symmetric function indexed by self.

INPUT:

• self – an element \(w\) of a Weyl group

Returns the affine Stanley symmetric function indexed by \(w\). Stanley symmetric functions are defined as generating series of the factorizations of \(w\) into Pieri factors and weighted by a statistic on Pieri factors.

See also

• stanley_symmetric_function_as_polynomial()

• WeylGroups.ParentMethods.pieri_factors()

• sage.combinat.root_system.pieri_factors

EXAMPLES:

Sage

sage: W = WeylGroup(['A', 3, 1])
sage: W.from_reduced_word([3,1,2,0,3,1,0]).stanley_symmetric_function()
8*m[1, 1, 1, 1, 1, 1, 1] + 4*m[2, 1, 1, 1, 1, 1]
+ 2*m[2, 2, 1, 1, 1] + m[2, 2, 2, 1]
sage: A = AffinePermutationGroup(['A',3,1])
sage: A.from_reduced_word([3,1,2,0,3,1,0]).stanley_symmetric_function()
8*m[1, 1, 1, 1, 1, 1, 1] + 4*m[2, 1, 1, 1, 1, 1]
+ 2*m[2, 2, 1, 1, 1] + m[2, 2, 2, 1]

sage: W = WeylGroup(['C',3,1])
sage: W.from_reduced_word([0,2,1,0]).stanley_symmetric_function()
32*m[1, 1, 1, 1] + 16*m[2, 1, 1] + 8*m[2, 2] + 4*m[3, 1]

sage: W = WeylGroup(['B',3,1])
sage: W.from_reduced_word([3,2,1]).stanley_symmetric_function()
2*m[1, 1, 1] + m[2, 1] + 1/2*m[3]

sage: W = WeylGroup(['B',4])
sage: w = W.from_reduced_word([3,2,3,1])
sage: w.stanley_symmetric_function()  # long time (6s on sage.math, 2011)
48*m[1, 1, 1, 1] + 24*m[2, 1, 1] + 12*m[2, 2] + 8*m[3, 1] + 2*m[4]

sage: A = AffinePermutationGroup(['A',4,1])
sage: a = A([-2,0,1,4,12])
sage: a.stanley_symmetric_function()
6*m[1, 1, 1, 1, 1, 1, 1, 1] + 5*m[2, 1, 1, 1, 1, 1, 1]
+ 4*m[2, 2, 1, 1, 1, 1] + 3*m[2, 2, 2, 1, 1] + 2*m[2, 2, 2, 2]
+ 4*m[3, 1, 1, 1, 1, 1] + 3*m[3, 2, 1, 1, 1] + 2*m[3, 2, 2, 1]
+ 2*m[3, 3, 1, 1] + m[3, 3, 2] + 3*m[4, 1, 1, 1, 1]
+ 2*m[4, 2, 1, 1] + m[4, 2, 2] + m[4, 3, 1]

Python

>>> from sage.all import *
>>> W = WeylGroup(['A', Integer(3), Integer(1)])
>>> W.from_reduced_word([Integer(3),Integer(1),Integer(2),Integer(0),Integer(3),Integer(1),Integer(0)]).stanley_symmetric_function()
8*m[1, 1, 1, 1, 1, 1, 1] + 4*m[2, 1, 1, 1, 1, 1]
+ 2*m[2, 2, 1, 1, 1] + m[2, 2, 2, 1]
>>> A = AffinePermutationGroup(['A',Integer(3),Integer(1)])
>>> A.from_reduced_word([Integer(3),Integer(1),Integer(2),Integer(0),Integer(3),Integer(1),Integer(0)]).stanley_symmetric_function()
8*m[1, 1, 1, 1, 1, 1, 1] + 4*m[2, 1, 1, 1, 1, 1]
+ 2*m[2, 2, 1, 1, 1] + m[2, 2, 2, 1]

>>> W = WeylGroup(['C',Integer(3),Integer(1)])
>>> W.from_reduced_word([Integer(0),Integer(2),Integer(1),Integer(0)]).stanley_symmetric_function()
32*m[1, 1, 1, 1] + 16*m[2, 1, 1] + 8*m[2, 2] + 4*m[3, 1]

>>> W = WeylGroup(['B',Integer(3),Integer(1)])
>>> W.from_reduced_word([Integer(3),Integer(2),Integer(1)]).stanley_symmetric_function()
2*m[1, 1, 1] + m[2, 1] + 1/2*m[3]

>>> W = WeylGroup(['B',Integer(4)])
>>> w = W.from_reduced_word([Integer(3),Integer(2),Integer(3),Integer(1)])
>>> w.stanley_symmetric_function()  # long time (6s on sage.math, 2011)
48*m[1, 1, 1, 1] + 24*m[2, 1, 1] + 12*m[2, 2] + 8*m[3, 1] + 2*m[4]

>>> A = AffinePermutationGroup(['A',Integer(4),Integer(1)])
>>> a = A([-Integer(2),Integer(0),Integer(1),Integer(4),Integer(12)])
>>> a.stanley_symmetric_function()
6*m[1, 1, 1, 1, 1, 1, 1, 1] + 5*m[2, 1, 1, 1, 1, 1, 1]
+ 4*m[2, 2, 1, 1, 1, 1] + 3*m[2, 2, 2, 1, 1] + 2*m[2, 2, 2, 2]
+ 4*m[3, 1, 1, 1, 1, 1] + 3*m[3, 2, 1, 1, 1] + 2*m[3, 2, 2, 1]
+ 2*m[3, 3, 1, 1] + m[3, 3, 2] + 3*m[4, 1, 1, 1, 1]
+ 2*m[4, 2, 1, 1] + m[4, 2, 2] + m[4, 3, 1]

Sage Live

W = WeylGroup(['A', 3, 1])
W.from_reduced_word([3,1,2,0,3,1,0]).stanley_symmetric_function()
A = AffinePermutationGroup(['A',3,1])
A.from_reduced_word([3,1,2,0,3,1,0]).stanley_symmetric_function()
W = WeylGroup(['C',3,1])
W.from_reduced_word([0,2,1,0]).stanley_symmetric_function()
W = WeylGroup(['B',3,1])
W.from_reduced_word([3,2,1]).stanley_symmetric_function()
W = WeylGroup(['B',4])
w = W.from_reduced_word([3,2,3,1])
w.stanley_symmetric_function()  # long time (6s on sage.math, 2011)
A = AffinePermutationGroup(['A',4,1])
a = A([-2,0,1,4,12])
a.stanley_symmetric_function()

One more example (Issue #14095):

Sage

sage: G = SymmetricGroup(4)
sage: w = G.from_reduced_word([3,2,3,1])
sage: w.stanley_symmetric_function()
3*m[1, 1, 1, 1] + 2*m[2, 1, 1] + m[2, 2] + m[3, 1]

Python

>>> from sage.all import *
>>> G = SymmetricGroup(Integer(4))
>>> w = G.from_reduced_word([Integer(3),Integer(2),Integer(3),Integer(1)])
>>> w.stanley_symmetric_function()
3*m[1, 1, 1, 1] + 2*m[2, 1, 1] + m[2, 2] + m[3, 1]

Sage Live

G = SymmetricGroup(4)
w = G.from_reduced_word([3,2,3,1])
w.stanley_symmetric_function()

REFERENCES:

• [BH1994]

• [Lam2008]

• [LSS2009]

• [Pon2010]

stanley_symmetric_function_as_polynomial(max_length=None)

Return a multivariate generating function for the number of factorizations of a Weyl group element into Pieri factors of decreasing length, weighted by a statistic on Pieri factors.

See also

• stanley_symmetric_function()

• WeylGroups.ParentMethods.pieri_factors()

• sage.combinat.root_system.pieri_factors

INPUT:

• self – an element \(w\) of a Weyl group \(W\)

• max_length – nonnegative integer or infinity (default: infinity)

Returns the generating series for the Pieri factorizations \(w = u_1 \cdots u_k\), where \(u_i\) is a Pieri factor for all \(i\), \(l(w) = \sum_{i=1}^k l(u_i)\) and max_length \(\geq l(u_1) \geq \cdots \geq l(u_k)\).

A factorization \(u_1 \cdots u_k\) contributes a monomial of the form \(\prod_i x_{l(u_i)}\), with coefficient given by \(\prod_i 2^{c(u_i)}\), where \(c\) is a type-dependent statistic on Pieri factors, as returned by the method u[i].stanley_symm_poly_weight().

EXAMPLES:

Sage

sage: W = WeylGroup(['A', 3, 1])
sage: W.from_reduced_word([]).stanley_symmetric_function_as_polynomial()
1
sage: W.from_reduced_word([1]).stanley_symmetric_function_as_polynomial()
x1
sage: W.from_reduced_word([1,2]).stanley_symmetric_function_as_polynomial()
x1^2
sage: W.from_reduced_word([2,1]).stanley_symmetric_function_as_polynomial()
x1^2 + x2
sage: W.from_reduced_word([1,2,1]).stanley_symmetric_function_as_polynomial()
2*x1^3 + x1*x2
sage: W.from_reduced_word([1,2,1,0]).stanley_symmetric_function_as_polynomial()
3*x1^4 + 2*x1^2*x2 + x2^2 + x1*x3
sage: x = W.from_reduced_word([1,2,3,1,2,1,0])
sage: x.stanley_symmetric_function_as_polynomial()  # long time
22*x1^7 + 11*x1^5*x2 + 5*x1^3*x2^2 + 3*x1^4*x3 + 2*x1*x2^3 + x1^2*x2*x3
sage: y = W.from_reduced_word([3,1,2,0,3,1,0])
sage: y.stanley_symmetric_function_as_polynomial()  # long time
8*x1^7 + 4*x1^5*x2 + 2*x1^3*x2^2 + x1*x2^3

sage: W = WeylGroup(['C',3,1])
sage: W.from_reduced_word([0,2,1,0]).stanley_symmetric_function_as_polynomial()
32*x1^4 + 16*x1^2*x2 + 8*x2^2 + 4*x1*x3

sage: W = WeylGroup(['B',3,1])
sage: W.from_reduced_word([3,2,1]).stanley_symmetric_function_as_polynomial()
2*x1^3 + x1*x2 + 1/2*x3

Python

>>> from sage.all import *
>>> W = WeylGroup(['A', Integer(3), Integer(1)])
>>> W.from_reduced_word([]).stanley_symmetric_function_as_polynomial()
1
>>> W.from_reduced_word([Integer(1)]).stanley_symmetric_function_as_polynomial()
x1
>>> W.from_reduced_word([Integer(1),Integer(2)]).stanley_symmetric_function_as_polynomial()
x1^2
>>> W.from_reduced_word([Integer(2),Integer(1)]).stanley_symmetric_function_as_polynomial()
x1^2 + x2
>>> W.from_reduced_word([Integer(1),Integer(2),Integer(1)]).stanley_symmetric_function_as_polynomial()
2*x1^3 + x1*x2
>>> W.from_reduced_word([Integer(1),Integer(2),Integer(1),Integer(0)]).stanley_symmetric_function_as_polynomial()
3*x1^4 + 2*x1^2*x2 + x2^2 + x1*x3
>>> x = W.from_reduced_word([Integer(1),Integer(2),Integer(3),Integer(1),Integer(2),Integer(1),Integer(0)])
>>> x.stanley_symmetric_function_as_polynomial()  # long time
22*x1^7 + 11*x1^5*x2 + 5*x1^3*x2^2 + 3*x1^4*x3 + 2*x1*x2^3 + x1^2*x2*x3
>>> y = W.from_reduced_word([Integer(3),Integer(1),Integer(2),Integer(0),Integer(3),Integer(1),Integer(0)])
>>> y.stanley_symmetric_function_as_polynomial()  # long time
8*x1^7 + 4*x1^5*x2 + 2*x1^3*x2^2 + x1*x2^3

>>> W = WeylGroup(['C',Integer(3),Integer(1)])
>>> W.from_reduced_word([Integer(0),Integer(2),Integer(1),Integer(0)]).stanley_symmetric_function_as_polynomial()
32*x1^4 + 16*x1^2*x2 + 8*x2^2 + 4*x1*x3

>>> W = WeylGroup(['B',Integer(3),Integer(1)])
>>> W.from_reduced_word([Integer(3),Integer(2),Integer(1)]).stanley_symmetric_function_as_polynomial()
2*x1^3 + x1*x2 + 1/2*x3

Sage Live

W = WeylGroup(['A', 3, 1])
W.from_reduced_word([]).stanley_symmetric_function_as_polynomial()
W.from_reduced_word([1]).stanley_symmetric_function_as_polynomial()
W.from_reduced_word([1,2]).stanley_symmetric_function_as_polynomial()
W.from_reduced_word([2,1]).stanley_symmetric_function_as_polynomial()
W.from_reduced_word([1,2,1]).stanley_symmetric_function_as_polynomial()
W.from_reduced_word([1,2,1,0]).stanley_symmetric_function_as_polynomial()
x = W.from_reduced_word([1,2,3,1,2,1,0])
x.stanley_symmetric_function_as_polynomial()  # long time
y = W.from_reduced_word([3,1,2,0,3,1,0])
y.stanley_symmetric_function_as_polynomial()  # long time
W = WeylGroup(['C',3,1])
W.from_reduced_word([0,2,1,0]).stanley_symmetric_function_as_polynomial()
W = WeylGroup(['B',3,1])
W.from_reduced_word([3,2,1]).stanley_symmetric_function_as_polynomial()

Algorithm: Induction on the left Pieri factors. Note that this induction preserves subsets of \(W\) which are stable by taking right factors, and in particular Grassmanian elements.

Finite

alias of FiniteWeylGroups

class ParentMethods

Bases: object

bruhat_cone(x, y, side='upper', backend='cdd')

Return the (upper or lower) Bruhat cone associated to the interval [x,y].

To a cover relation \(v \prec w\) in strong Bruhat order you can assign a positive root \(\beta\) given by the unique reflection \(s_\beta\) such that \(s_\beta v = w\).

The upper Bruhat cone of the interval \([x,y]\) is the non-empty, polyhedral cone generated by the roots corresponding to \(x \prec a\) for all atoms \(a\) in the interval. The lower Bruhat cone of the interval \([x,y]\) is the non-empty, polyhedral cone generated by the roots corresponding to \(c \prec y\) for all coatoms \(c\) in the interval.

INPUT:

• x – an element in the group \(W\)

• y – an element in the group \(W\)

• side – (default: 'upper') must be one of the following:

  • 'upper' – return the upper Bruhat cone of the interval [x, y]

  • 'lower' – return the lower Bruhat cone of the interval [x, y]

• backend – string (default: 'cdd'); the backend to use to create the polyhedron

EXAMPLES:

Sage

sage: W = WeylGroup(['A',2])
sage: x = W.from_reduced_word([1])
sage: y = W.w0
sage: W.bruhat_cone(x, y)
A 2-dimensional polyhedron in QQ^3
 defined as the convex hull of 1 vertex and 2 rays

sage: W = WeylGroup(['E',6])
sage: x = W.one()
sage: y = W.w0
sage: W.bruhat_cone(x, y, side='lower')
A 6-dimensional polyhedron in QQ^8
 defined as the convex hull of 1 vertex and 6 rays

Python

>>> from sage.all import *
>>> W = WeylGroup(['A',Integer(2)])
>>> x = W.from_reduced_word([Integer(1)])
>>> y = W.w0
>>> W.bruhat_cone(x, y)
A 2-dimensional polyhedron in QQ^3
 defined as the convex hull of 1 vertex and 2 rays

>>> W = WeylGroup(['E',Integer(6)])
>>> x = W.one()
>>> y = W.w0
>>> W.bruhat_cone(x, y, side='lower')
A 6-dimensional polyhedron in QQ^8
 defined as the convex hull of 1 vertex and 6 rays

Sage Live

W = WeylGroup(['A',2])
x = W.from_reduced_word([1])
y = W.w0
W.bruhat_cone(x, y)
W = WeylGroup(['E',6])
x = W.one()
y = W.w0
W.bruhat_cone(x, y, side='lower')

REFERENCES:

• [Dy1994]

• [JS2021]

coxeter_matrix()

Return the Coxeter matrix associated to self.

EXAMPLES:

Sage

sage: G = WeylGroup(['A',3])
sage: G.coxeter_matrix()
[1 3 2]
[3 1 3]
[2 3 1]

Python

>>> from sage.all import *
>>> G = WeylGroup(['A',Integer(3)])
>>> G.coxeter_matrix()
[1 3 2]
[3 1 3]
[2 3 1]

Sage Live

G = WeylGroup(['A',3])
G.coxeter_matrix()

pieri_factors(*args, **keywords)

Return the set of Pieri factors in this Weyl group.

For any type, the set of Pieri factors forms a lower ideal in Bruhat order, generated by all the conjugates of some special element of the Weyl group. In type \(A_n\), this special element is \(s_n\cdots s_1\), and the conjugates are obtained by rotating around this reduced word.

These are used to compute Stanley symmetric functions.

See also

• WeylGroups.ElementMethods.stanley_symmetric_function()

• sage.combinat.root_system.pieri_factors

EXAMPLES:

Sage

sage: W = WeylGroup(['A',5,1])
sage: PF = W.pieri_factors()
sage: PF.cardinality()
63

sage: W = WeylGroup(['B',3])
sage: PF = W.pieri_factors()
sage: sorted([w.reduced_word() for w in PF])
[[],
 [1],
 [1, 2],
 [1, 2, 1],
 [1, 2, 3],
 [1, 2, 3, 1],
 [1, 2, 3, 2],
 [1, 2, 3, 2, 1],
 [2],
 [2, 1],
 [2, 3],
 [2, 3, 1],
 [2, 3, 2],
 [2, 3, 2, 1],
 [3],
 [3, 1],
 [3, 1, 2],
 [3, 1, 2, 1],
 [3, 2],
 [3, 2, 1]]
sage: W = WeylGroup(['C',4,1])
sage: PF = W.pieri_factors()
sage: W.from_reduced_word([3,2,0]) in PF
True

Python

>>> from sage.all import *
>>> W = WeylGroup(['A',Integer(5),Integer(1)])
>>> PF = W.pieri_factors()
>>> PF.cardinality()
63

>>> W = WeylGroup(['B',Integer(3)])
>>> PF = W.pieri_factors()
>>> sorted([w.reduced_word() for w in PF])
[[],
 [1],
 [1, 2],
 [1, 2, 1],
 [1, 2, 3],
 [1, 2, 3, 1],
 [1, 2, 3, 2],
 [1, 2, 3, 2, 1],
 [2],
 [2, 1],
 [2, 3],
 [2, 3, 1],
 [2, 3, 2],
 [2, 3, 2, 1],
 [3],
 [3, 1],
 [3, 1, 2],
 [3, 1, 2, 1],
 [3, 2],
 [3, 2, 1]]
>>> W = WeylGroup(['C',Integer(4),Integer(1)])
>>> PF = W.pieri_factors()
>>> W.from_reduced_word([Integer(3),Integer(2),Integer(0)]) in PF
True

Sage Live

W = WeylGroup(['A',5,1])
PF = W.pieri_factors()
PF.cardinality()
W = WeylGroup(['B',3])
PF = W.pieri_factors()
sorted([w.reduced_word() for w in PF])
W = WeylGroup(['C',4,1])
PF = W.pieri_factors()
W.from_reduced_word([3,2,0]) in PF

quantum_bruhat_graph(index_set=())

Return the quantum Bruhat graph of the quotient of the Weyl group by a parabolic subgroup \(W_J\).

INPUT:

• index_set – (default: ()) a tuple \(J\) of nodes of the Dynkin diagram

By default, the value for index_set indicates that the subgroup is trivial and the quotient is the full Weyl group.

EXAMPLES:

Sage

sage: W = WeylGroup(['A',3], prefix='s')
sage: g = W.quantum_bruhat_graph((1,3))
sage: g
Parabolic Quantum Bruhat Graph of Weyl Group of type ['A', 3]
 (as a matrix group acting on the ambient space)
 for nodes (1, 3): Digraph on 6 vertices
sage: g.vertices(sort=True)
[s2*s3*s1*s2, s3*s1*s2, s1*s2, s3*s2, s2, 1]
sage: g.edges(sort=True)
[(s2*s3*s1*s2, s2, alpha[2]),
 (s3*s1*s2, s2*s3*s1*s2, alpha[1] + alpha[2] + alpha[3]),
 (s3*s1*s2, 1, alpha[2]),
 (s1*s2, s3*s1*s2, alpha[2] + alpha[3]),
 (s3*s2, s3*s1*s2, alpha[1] + alpha[2]),
 (s2, s1*s2, alpha[1] + alpha[2]),
 (s2, s3*s2, alpha[2] + alpha[3]),
 (1, s2, alpha[2])]
sage: W = WeylGroup(['A',3,1], prefix='s')
sage: g = W.quantum_bruhat_graph()
Traceback (most recent call last):
...
ValueError: the Cartan type ['A', 3, 1] is not finite

Python

>>> from sage.all import *
>>> W = WeylGroup(['A',Integer(3)], prefix='s')
>>> g = W.quantum_bruhat_graph((Integer(1),Integer(3)))
>>> g
Parabolic Quantum Bruhat Graph of Weyl Group of type ['A', 3]
 (as a matrix group acting on the ambient space)
 for nodes (1, 3): Digraph on 6 vertices
>>> g.vertices(sort=True)
[s2*s3*s1*s2, s3*s1*s2, s1*s2, s3*s2, s2, 1]
>>> g.edges(sort=True)
[(s2*s3*s1*s2, s2, alpha[2]),
 (s3*s1*s2, s2*s3*s1*s2, alpha[1] + alpha[2] + alpha[3]),
 (s3*s1*s2, 1, alpha[2]),
 (s1*s2, s3*s1*s2, alpha[2] + alpha[3]),
 (s3*s2, s3*s1*s2, alpha[1] + alpha[2]),
 (s2, s1*s2, alpha[1] + alpha[2]),
 (s2, s3*s2, alpha[2] + alpha[3]),
 (1, s2, alpha[2])]
>>> W = WeylGroup(['A',Integer(3),Integer(1)], prefix='s')
>>> g = W.quantum_bruhat_graph()
Traceback (most recent call last):
...
ValueError: the Cartan type ['A', 3, 1] is not finite

Sage Live

W = WeylGroup(['A',3], prefix='s')
g = W.quantum_bruhat_graph((1,3))
g
g.vertices(sort=True)
g.edges(sort=True)
W = WeylGroup(['A',3,1], prefix='s')
g = W.quantum_bruhat_graph()

additional_structure()

Return None.

Indeed, the category of Weyl groups defines no additional structure: Weyl groups are a special class of Coxeter groups.

See also

Category.additional_structure()

Todo

Should this category be a CategoryWithAxiom?

EXAMPLES:

Sage

sage: WeylGroups().additional_structure()

Python

>>> from sage.all import *
>>> WeylGroups().additional_structure()

Sage Live

WeylGroups().additional_structure()

super_categories()

EXAMPLES:

Sage

sage: WeylGroups().super_categories()
[Category of Coxeter groups]

Python

>>> from sage.all import *
>>> WeylGroups().super_categories()
[Category of Coxeter groups]

Sage Live

WeylGroups().super_categories()

Make live