Weyl Groups¶
- class sage.categories.weyl_groups.WeylGroups[source]¶
- 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] - >>> from sage.all import * >>> WeylGroups() Category of Weyl groups >>> 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) - >>> 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) - This one will eventually be also in this category: - sage: SymmetricGroup(4) Symmetric group of order 4! as a permutation group - >>> from sage.all import * >>> SymmetricGroup(Integer(4)) Symmetric group of order 4! as a permutation group - class ElementMethods[source]¶
- Bases: - object- bruhat_lower_covers_coroots()[source]¶
- Return all 2-tuples ( - v, \(\alpha\)) where- vis covered by- selfand \(\alpha\) 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])] - >>> 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])] 
 - bruhat_upper_covers_coroots()[source]¶
- Return all 2-tuples ( - v, \(\alpha\)) where- vis covers- selfand \(\alpha\) 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])] - >>> 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])] 
 - inversion_arrangement(side='right')[source]¶
- 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 - selfgiven as roots.- The - sideparameter 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) - >>> 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) - The identity element gives the empty arrangement: - sage: W = WeylGroup(['A',3]) sage: W.one().inversion_arrangement() Empty hyperplane arrangement of dimension 3 - >>> from sage.all import * >>> W = WeylGroup(['A',Integer(3)]) >>> W.one().inversion_arrangement() Empty hyperplane arrangement of dimension 3 
 - inversions(side='right', inversion_type='reflections')[source]¶
- 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- selfto negative (co)roots; their associated reflections are described above.- If - sideis ‘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]] - >>> 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]] 
 - is_pieri_factor()[source]¶
- Return whether - selfis a Pieri factor, as used for computing Stanley symmetric functions.- 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 - >>> 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 
 - left_pieri_factorizations(max_length=None)[source]¶
- Return all factorizations of - selfas \(uv\), where \(u\) is a Pieri factor and \(v\) is an element of the Weyl group.- 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 - >>> 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 
 - quantum_bruhat_successors(index_set=None, roots=False, quantum_only=False)[source]¶
- Return the successors of - selfin 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- wis 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: 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) - >>> 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) 
 - reflection_to_coroot()[source]¶
- 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 - >>> 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 
 - reflection_to_root()[source]¶
- 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 - >>> 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 
 - stanley_symmetric_function()[source]¶
- 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] - >>> 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] - One more example (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] - >>> 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] - REFERENCES: 
 - stanley_symmetric_function_as_polynomial(max_length=None)[source]¶
- 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 - 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: 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 - >>> 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 - 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[source]¶
- alias of - FiniteWeylGroups
 - class ParentMethods[source]¶
- Bases: - object- bruhat_cone(x, y, side='upper', backend='cdd')[source]¶
- 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 - >>> 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 - REFERENCES: 
 - coxeter_matrix()[source]¶
- 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] - >>> from sage.all import * >>> G = WeylGroup(['A',Integer(3)]) >>> G.coxeter_matrix() [1 3 2] [3 1 3] [2 3 1] 
 - pieri_factors(*args, **keywords)[source]¶
- 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 - 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 - >>> 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 
 - quantum_bruhat_graph(index_set=())[source]¶
- 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_setindicates 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 - >>> 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 
 
 - additional_structure()[source]¶
- Return - None.- Indeed, the category of Weyl groups defines no additional structure: Weyl groups are a special class of Coxeter groups. - See also - Todo - Should this category be a - CategoryWithAxiom?- EXAMPLES: - sage: WeylGroups().additional_structure() - >>> from sage.all import * >>> WeylGroups().additional_structure()