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

Here are some examples:

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)

This one will eventually be also in this category:

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

Returns 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: 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])]
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: 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)

The identity element gives the empty arrangement:

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

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

See also

EXAMPLES:

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
left_pieri_factorizations(max_length=None)#

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

See also

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: 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
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 – (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 – (default: False) if True, returns only the quantum successors

EXAMPLES:

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)
reflection_to_coroot()#

Return the coroot associated with the reflection self.

EXAMPLES:

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
reflection_to_root()#

Return the root associated with the reflection self.

EXAMPLES:

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
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

EXAMPLES:

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]

One more example (github issue #14095):

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]

REFERENCES:

stanley_symmetric_function_as_polynomial(max_length=None)#

Returns 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

INPUT:

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

  • max_length – a non negative 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: 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

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: 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

REFERENCES:

coxeter_matrix()#

Return the Coxeter matrix associated to self.

EXAMPLES:

sage: G = WeylGroup(['A',3])
sage: G.coxeter_matrix()
[1 3 2]
[3 1 3]
[2 3 1]
pieri_factors(*args, **keywords)#

Returns 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

EXAMPLES:

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
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: 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
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: WeylGroups().additional_structure()
super_categories()#

EXAMPLES:

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