Coxeter Groups

class sage.categories.coxeter_groups.CoxeterGroups(s=None)

Bases: sage.categories.category_singleton.Category_singleton

The category of Coxeter groups.

A Coxeter group is a group \(W\) with a distinguished (finite) family of involutions \((s_i)_{i\in I}\), called the simple reflections, subject to relations of the form \((s_is_j)^{m_{i,j}} = 1\).

\(I\) is the index set of \(W\) and \(|I|\) is the rank of \(W\).

See http://en.wikipedia.org/wiki/Coxeter_group for details.

EXAMPLES:

sage: C = CoxeterGroups(); C
Category of coxeter groups
sage: C.super_categories()
[Category of groups, Category of enumerated sets]

sage: W = C.example(); W
The symmetric group on {0, ..., 3}

sage: W.simple_reflections()
Finite family {0: (1, 0, 2, 3), 1: (0, 2, 1, 3), 2: (0, 1, 3, 2)}

Here are some further examples:

sage: FiniteCoxeterGroups().example()
The 5-th dihedral group of order 10
sage: FiniteWeylGroups().example()
The symmetric group on {0, ..., 3}
sage: WeylGroup(["B", 3])
Weyl Group of type ['B', 3] (as a matrix group acting on the ambient space)

Those will eventually be also in this category:

sage: SymmetricGroup(4)
Symmetric group of order 4! as a permutation group
sage: DihedralGroup(5)
Dihedral group of order 10 as a permutation group

Todo

add a demo of usual computations on Coxeter groups.

See also

WeylGroups, sage.combinat.root_system

TESTS:

sage: W = CoxeterGroups().example(); TestSuite(W).run(verbose = "True")
running ._test_an_element() . . . pass
running ._test_associativity() . . . pass
running ._test_category() . . . pass
running ._test_elements() . . .
  Running the test suite of self.an_element()
  running ._test_category() . . . pass
  running ._test_eq() . . . pass
  running ._test_not_implemented_methods() . . . pass
  running ._test_pickling() . . . pass
  pass
running ._test_elements_eq_reflexive() . . . pass
running ._test_elements_eq_symmetric() . . . pass
running ._test_elements_eq_transitive() . . . pass
running ._test_elements_neq() . . . pass
running ._test_enumerated_set_contains() . . . pass
running ._test_enumerated_set_iter_cardinality() . . . pass
running ._test_enumerated_set_iter_list() . . . pass
running ._test_eq() . . . pass
running ._test_has_descent() . . . pass
running ._test_inverse() . . . pass
running ._test_not_implemented_methods() . . . pass
running ._test_one() . . . pass
running ._test_pickling() . . . pass
running ._test_prod() . . . pass
running ._test_reduced_word() . . . pass
running ._test_simple_projections() . . . pass
running ._test_some_elements() . . . pass
class ElementMethods
absolute_le(other)

Return whether self is smaller than other in the absolute order.

A general reflection is an element of the form \(w s_i w^{-1}\), where \(s_i\) is a simple reflection. The absolute order is defined analogously to the weak order but using general reflections rather than just simple reflections.

This partial order can be used to define noncrossing partitions associated with this Coxeter group.

EXAMPLES:

sage: W = WeylGroup(["A", 3])
sage: s = W.simple_reflections()
sage: w0 = s[1]
sage: w1 = s[1]*s[2]*s[3]
sage: w0.absolute_le(w1)
True
sage: w1.absolute_le(w0)
False
sage: w1.absolute_le(w1)
True
absolute_length()

Return the absolute length of self

The absolute length is the length of the shortest expression of the element as a product of reflections.

See also

absolute_le().

EXAMPLES:

sage: W = WeylGroup(["A", 3])
sage: s = W.simple_reflections()
sage: (s[1]*s[2]*s[3]).absolute_length()
3
apply_conjugation_by_simple_reflection(i)

Conjugates self by the i-th simple reflection.

EXAMPLES:

sage: W = WeylGroup(['A',3])
sage: w = W.from_reduced_word([3,1,2,1])
sage: w.apply_conjugation_by_simple_reflection(1).reduced_word()
[3, 2]
apply_demazure_product(element, side='right', length_increasing=True)

Returns the Demazure or 0-Hecke product of self with another Coxeter group element.

See CoxeterGroups.ParentMethods.simple_projections().

INPUT:

  • element – either an element of the same Coxeter

    group as self or a tuple or a list (such as a reduced word) of elements from the index set of the Coxeter group.

  • side – ‘left’ or ‘right’ (default: ‘right’); the

    side of self on which the element should be applied. If side is ‘left’ then the operation is applied on the left.

  • length_increasing – a boolean (default True)

    whether to act length increasingly or decreasingly

EXAMPLES:

sage: W = WeylGroup(['C',4],prefix="s")
sage: v = W.from_reduced_word([1,2,3,4,3,1])
sage: v.apply_demazure_product([1,3,4,3,3])
s4*s1*s2*s3*s4*s3*s1
sage: v.apply_demazure_product([1,3,4,3],side='left')
s3*s4*s1*s2*s3*s4*s2*s3*s1
sage: v.apply_demazure_product((1,3,4,3),side='left')
s3*s4*s1*s2*s3*s4*s2*s3*s1
sage: v.apply_demazure_product(v)
s2*s3*s4*s1*s2*s3*s4*s2*s3*s2*s1
apply_simple_projection(i, side='right', length_increasing=True, toward_max=None)

INPUT:

  • i - an element of the index set of the Coxeter group
  • side - ‘left’ or ‘right’ (default: ‘right’)
  • length_increasing - a boolean (default: True) specifying the direction of the projection

Returns the result of the application of the simple projection \(\pi_i\) (resp. \(\overline\pi_i\)) on self.

See CoxeterGroups.ParentMethods.simple_projections() for the definition of the simple projections.

EXAMPLE:

sage: W=CoxeterGroups().example()
sage: w=W.an_element()
sage: w
(1, 2, 3, 0)
sage: w.apply_simple_projection(2)
(1, 2, 3, 0)
sage: w.apply_simple_projection(2, length_increasing=False)
(1, 2, 0, 3)
sage: W = WeylGroup(['C',4],prefix="s")
sage: v = W.from_reduced_word([1,2,3,4,3,1])
sage: v
s1*s2*s3*s4*s3*s1
sage: v.apply_simple_projection(2)
s1*s2*s3*s4*s3*s1*s2
sage: v.apply_simple_projection(2, side='left')
s1*s2*s3*s4*s3*s1
sage: v.apply_simple_projection(1, length_increasing = False)
s1*s2*s3*s4*s3
apply_simple_reflection(i, side='right')

Returns self multiplied by the simple reflection s[i]

INPUT:

  • i – an element of the index set
  • side – “left” or “right” (default: “right”)

This default implementation simply calls apply_simple_reflection_left() or apply_simple_reflection_right().

EXAMPLES:

sage: W=CoxeterGroups().example()
sage: w = W.an_element(); w
(1, 2, 3, 0)
sage: w.apply_simple_reflection(0, side = "left")
(0, 2, 3, 1)
sage: w.apply_simple_reflection(1, side = "left")
(2, 1, 3, 0)
sage: w.apply_simple_reflection(2, side = "left")
(1, 3, 2, 0)

sage: w.apply_simple_reflection(0, side = "right")
(2, 1, 3, 0)
sage: w.apply_simple_reflection(1, side = "right")
(1, 3, 2, 0)
sage: w.apply_simple_reflection(2, side = "right")
(1, 2, 0, 3)

By default, side is “right”:

sage: w.apply_simple_reflection(0)
(2, 1, 3, 0)

TESTS:

sage: w.apply_simple_reflection_right.__module__
'sage.categories.coxeter_groups'
apply_simple_reflection_left(i)

Returns self multiplied by the simple reflection s[i] on the left

This low level method is used intensively. Coxeter groups are encouraged to override this straightforward implementation whenever a faster approach exists.

EXAMPLES:

sage: W=CoxeterGroups().example()
sage: w = W.an_element(); w
(1, 2, 3, 0)
sage: w.apply_simple_reflection_left(0)
(0, 2, 3, 1)
sage: w.apply_simple_reflection_left(1)
(2, 1, 3, 0)
sage: w.apply_simple_reflection_left(2)
(1, 3, 2, 0)

TESTS:

sage: w.apply_simple_reflection_left.__module__
'sage.categories.coxeter_groups'
apply_simple_reflection_right(i)

Returns self multiplied by the simple reflection s[i] on the right

This low level method is used intensively. Coxeter groups are encouraged to override this straightforward implementation whenever a faster approach exists.

EXAMPLES:

sage: W=CoxeterGroups().example()
sage: w = W.an_element(); w
(1, 2, 3, 0)
sage: w.apply_simple_reflection_right(0)
(2, 1, 3, 0)
sage: w.apply_simple_reflection_right(1)
(1, 3, 2, 0)
sage: w.apply_simple_reflection_right(2)
(1, 2, 0, 3)

TESTS:

sage: w.apply_simple_reflection_right.__module__
'sage.categories.coxeter_groups'
apply_simple_reflections(word, side='right')

INPUT:

  • word – A sequence of indices of Coxeter generators
  • side – Indicates multiplying from left or right

Returns the result of the (left/right) multiplication of word to self. self is not changed.

EXAMPLES:

sage: W=CoxeterGroups().example()
sage: w=W.an_element(); w
(1, 2, 3, 0)
sage: w.apply_simple_reflections([0,1])
(2, 3, 1, 0)
sage: w
(1, 2, 3, 0)
sage: w.apply_simple_reflections([0,1],side='left')
(0, 1, 3, 2)
binary_factorizations(predicate=The constant function (...) -> True)

Returns the set of all the factorizations \(self = u v\) such that \(l(self) = l(u) + l(v)\).

Iterating through this set is Constant Amortized Time (counting arithmetic operations in the Coxeter group as constant time) complexity, and memory linear in the length of \(self\).

One can pass as optional argument a predicate p such that \(p(u)\) implies \(p(u')\) for any \(u\) left factor of \(self\) and \(u'\) left factor of \(u\). Then this returns only the factorizations \(self = uv\) such \(p(u)\) holds.

EXAMPLES:

We construct the set of all factorizations of the maximal element of the group:

sage: W = WeylGroup(['A',3])
sage: s = W.simple_reflections()
sage: w0 = W.from_reduced_word([1,2,3,1,2,1])
sage: w0.binary_factorizations().cardinality()
24

The same number of factorizations, by bounded length:

sage: [w0.binary_factorizations(lambda u: u.length() <= l).cardinality() for l in [-1,0,1,2,3,4,5,6]]
[0, 1, 4, 9, 15, 20, 23, 24]

The number of factorizations of the elements just below the maximal element:

sage: [(s[i]*w0).binary_factorizations().cardinality() for i in [1,2,3]]
[12, 12, 12]
sage: w0.binary_factorizations(lambda u: False).cardinality()
0

TESTS:

sage: w0.binary_factorizations().category()
Category of finite enumerated sets
bruhat_le(other)

Bruhat comparison

INPUT:

  • other - an element of the same Coxeter group

OUTPUT: a boolean

Returns whether self <= other in the Bruhat order.

EXAMPLES:

sage: W = WeylGroup(["A",3])
sage: u = W.from_reduced_word([1,2,1])
sage: v = W.from_reduced_word([1,2,3,2,1])
sage: u.bruhat_le(u)
True
sage: u.bruhat_le(v)
True
sage: v.bruhat_le(u)
False
sage: v.bruhat_le(v)
True
sage: s = W.simple_reflections()
sage: s[1].bruhat_le(W.one())
False

The implementation uses the equivalent condition that any reduced word for other contains a reduced word for self as subword. See Stembridge, A short derivation of the Mobius function for the Bruhat order. J. Algebraic Combin. 25 (2007), no. 2, 141–148, Proposition 1.1.

Complexity: \(O(l * c)\), where \(l\) is the minimum of the lengths of \(u\) and of \(v\), and \(c\) is the cost of the low level methods first_descent(), has_descent(), apply_simple_reflection(), etc. Those are typically \(O(n)\), where \(n\) is the rank of the Coxeter group.

TESTS:

We now run consistency tests with permutations and bruhat_lower_covers():

sage: W = WeylGroup(["A",3])
sage: P4 = Permutations(4)
sage: def P4toW(w): return W.from_reduced_word(w.reduced_word())
sage: for u in P4:
...       for v in P4:
...           assert u.bruhat_lequal(v) == P4toW(u).bruhat_le(P4toW(v))

sage: W = WeylGroup(["B",3])
sage: P = W.bruhat_poset() # This is built from bruhat_lower_covers
sage: Q = Poset((W, attrcall("bruhat_le")))                             # long time (10s)
sage: all( u.bruhat_le(v) == P.is_lequal(u,v) for u in W for v in W ) # long time  (7s)
True
sage: all( P.is_lequal(u,v) == Q.is_lequal(u,v) for u in W for v in W)       # long time  (9s)
True
bruhat_lower_covers()

Returns all elements that self covers in (strong) Bruhat order.

If w = self has a descent at \(i\), then the elements that \(w\) covers are exactly \(\{ws_i, u_1s_i, u_2s_i,..., u_js_i\}\), where the \(u_k\) are elements that \(ws_i\) covers that also do not have a descent at \(i\).

EXAMPLES:

sage: W = WeylGroup(["A",3])
sage: w = W.from_reduced_word([3,2,3])
sage: print([v.reduced_word() for v in w.bruhat_lower_covers()])
[[3, 2], [2, 3]]

sage: W = WeylGroup(["A",3])
sage: print([v.reduced_word() for v in W.simple_reflection(1).bruhat_lower_covers()])
[[]]
sage: print([v.reduced_word() for v in W.one().bruhat_lower_covers()])
[]
sage: W = WeylGroup(["B",4,1])
sage: w = W.from_reduced_word([0,2])
sage: print([v.reduced_word() for v in w.bruhat_lower_covers()])
[[2], [0]]

We now show how to construct the Bruhat poset:

sage: W = WeylGroup(["A",3])
sage: covers = tuple([u, v] for v in W for u in v.bruhat_lower_covers() )
sage: P = Poset((W, covers), cover_relations = True)
sage: P.show()

Alternatively, one can just use:

sage: P = W.bruhat_poset()

The algorithm is taken from Stembridge’s ‘coxeter/weyl’ package for Maple.

bruhat_lower_covers_reflections()

Returns all 2-tuples of lower_covers and reflections (v, r) where v is covered by self and r is the reflection such that self = v r.

ALGORITHM:

See bruhat_lower_covers()

EXAMPLES:

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

Returns all elements that cover self in (strong) Bruhat order.

The algorithm works recursively, using the ‘inverse’ of the method described for lower covers bruhat_lower_covers(). Namely, it runs through all \(i\) in the index set. Let \(w\) equal self. If \(w\) has no right descent \(i\), then \(w s_i\) is a cover; if \(w\) has a decent at \(i\), then \(u_j s_i\) is a cover of \(w\) where \(u_j\) is a cover of \(w s_i\).

EXAMPLES:

sage: W = WeylGroup(['A',3,1], prefix="s")
sage: w = W.from_reduced_word([1,2,1])
sage: w.bruhat_upper_covers()
[s1*s2*s1*s0, s1*s2*s0*s1, s0*s1*s2*s1, s3*s1*s2*s1, s2*s3*s1*s2, s1*s2*s3*s1]

sage: W = WeylGroup(['A',3])
sage: w = W.long_element()
sage: w.bruhat_upper_covers()
[]

sage: W = WeylGroup(['A',3])
sage: w = W.from_reduced_word([1,2,1])
sage: S = [v for v in W if w in v.bruhat_lower_covers()]
sage: C = w.bruhat_upper_covers()
sage: set(S) == set(C)
True
bruhat_upper_covers_reflections()

Returns all 2-tuples of covers and reflections (v, r) where v covers self and r is the reflection such that self = v r.

ALGORITHM:

See bruhat_upper_covers()

EXAMPLES:

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

Return the matrix of self in the canonical faithful representation.

This is an \(n\)-dimension real faithful essential representation, where \(n\) is the number of generators of the Coxeter group. Note that this is not always the most natural matrix representation, for instance in type \(A_n\).

EXAMPLES:

sage: W = WeylGroup(["A", 3])
sage: s = W.simple_reflections()
sage: (s[1]*s[2]*s[3]).canonical_matrix()
[ 0  0 -1]
[ 1  0 -1]
[ 0  1 -1]
coset_representative(index_set, side='right')

INPUT:

  • index_set - a subset (or iterable) of the nodes of the Dynkin diagram
  • side - ‘left’ or ‘right’

Returns the unique shortest element of the Coxeter group \(W\) which is in the same left (resp. right) coset as self, with respect to the parabolic subgroup \(W_I\).

EXAMPLES:

sage: W = CoxeterGroups().example(5)
sage: s = W.simple_reflections()
sage: w = s[2]*s[1]*s[3]
sage: w.coset_representative([]).reduced_word()
[2, 3, 1]
sage: w.coset_representative([1]).reduced_word()
[2, 3]
sage: w.coset_representative([1,2]).reduced_word()
[2, 3]
sage: w.coset_representative([1,3]                 ).reduced_word()
[2]
sage: w.coset_representative([2,3]                 ).reduced_word()
[2, 1]
sage: w.coset_representative([1,2,3]               ).reduced_word()
[]
sage: w.coset_representative([],      side = 'left').reduced_word()
[2, 3, 1]
sage: w.coset_representative([1],     side = 'left').reduced_word()
[2, 3, 1]
sage: w.coset_representative([1,2],   side = 'left').reduced_word()
[3]
sage: w.coset_representative([1,3],   side = 'left').reduced_word()
[2, 3, 1]
sage: w.coset_representative([2,3],   side = 'left').reduced_word()
[1]
sage: w.coset_representative([1,2,3], side = 'left').reduced_word()
[]
cover_reflections(side='right')

Returns the set of reflections t such that self t covers self.

If side is ‘left’, t self covers self.

EXAMPLES:

sage: W = WeylGroup(['A',4], prefix="s")
sage: w = W.from_reduced_word([3,1,2,1])
sage: w.cover_reflections()
[s3, s2*s3*s2, s4, s1*s2*s3*s4*s3*s2*s1]
sage: w.cover_reflections(side = 'left')
[s4, s2, s1*s2*s1, s3*s4*s3]
deodhar_factor_element(w, index_set)

Returns Deodhar’s Bruhat order factoring element.

INPUT:

  • w is an element of the same Coxeter group W as self
  • index_set is a subset of Dynkin nodes defining a parabolic subgroup W' of W

It is assumed that v = self and w are minimum length coset representatives for W/W' such that v \(\le\) w in Bruhat order.

OUTPUT:

Deodhar’s element f(v,w) is the unique element of W' such that, for all v' and w' in W', vv' \(\le\) ww' in W if and only if v' \(\le\) f(v,w) * w' in W' where * is the Demazure product.

EXAMPLES:

sage: W = WeylGroup(['A',5],prefix="s")
sage: v = W.from_reduced_word([5])
sage: w = W.from_reduced_word([4,5,2,3,1,2])
sage: v.deodhar_factor_element(w,[1,3,4])
s3*s1
sage: W=WeylGroup(['C',2])
sage: w=W.from_reduced_word([2,1])
sage: w.deodhar_factor_element(W.from_reduced_word([2]),[1])
Traceback (most recent call last):
...
ValueError: [2, 1] is not of minimum length in its coset for the parabolic subgroup with index set [1]

REFERENCES:

[Deodhar]
  1. Deodhar, A splitting criterion for the Bruhat orderings on Coxeter groups. Comm. Algebra, 15:1889-1894, 1987.
deodhar_lift_down(w, index_set)

Letting v = self, given a Bruhat relation v W' \(\ge\) w W' among cosets with respect to the subgroup W' given by the Dynkin node subset index_set, returns the Bruhat-maximum lift x of wW' such that v \(\ge\) x.

INPUT:

  • w is an element of the same Coxeter group W as self.
  • index_set is a subset of Dynkin nodes defining a parabolic subgroup W'.

OUTPUT:

The unique Bruhat-maximum element x in W such that x W' = w W' and v `\ge` ``x.

EXAMPLES:

sage: W = WeylGroup(['A',3],prefix="s")
sage: v = W.from_reduced_word([1,2,3,2])
sage: w = W.from_reduced_word([3,2])
sage: v.deodhar_lift_down(w, [3])
s2*s3*s2
deodhar_lift_up(w, index_set)

Letting v = self, given a Bruhat relation v W' \(\le\) w W' among cosets with respect to the subgroup W' given by the Dynkin node subset index_set, returns the Bruhat-minimum lift x of wW' such that v \(\le\) x.

INPUT:

  • w is an element of the same Coxeter group W as self.
  • index_set is a subset of Dynkin nodes defining a parabolic subgroup W'.

OUTPUT:

The unique Bruhat-minimum element x in W such that x W' = w W' and v \(\le\) x.

EXAMPLES:

sage: W = WeylGroup(['A',3],prefix="s")
sage: v = W.from_reduced_word([1,2,3])
sage: w = W.from_reduced_word([1,3,2])
sage: v.deodhar_lift_up(w, [3])
s1*s2*s3*s2
descents(side='right', index_set=None, positive=False)

INPUT:

  • index_set - a subset (as a list or iterable) of the nodes of the Dynkin diagram; (default: all of them)
  • side - ‘left’ or ‘right’ (default: ‘right’)
  • positive - a boolean (default: False)

Returns the descents of self, as a list of elements of the index_set.

The index_set option can be used to restrict to the parabolic subgroup indexed by index_set.

If positive is True, then returns the non-descents instead

TODO: find a better name for positive: complement? non_descent?

Caveat: the return type may change to some other iterable (tuple, ...) in the future. Please use keyword arguments also, as the order of the arguments may change as well.

EXAMPLES:

sage: W=CoxeterGroups().example()
sage: s=W.simple_reflections()
sage: w=s[0]*s[1]
sage: w.descents()
[1]
sage: w=s[0]*s[2]
sage: w.descents()
[0, 2]

TODO: side, index_set, positive
first_descent(side='right', index_set=None, positive=False)

Returns the first left (resp. right) descent of self, as ane element of index_set, or None if there is none.

See descents() for a description of the options.

EXAMPLES:

sage: W = CoxeterGroups().example()
sage: s = W.simple_reflections()
sage: w = s[2]*s[0]
sage: w.first_descent()
0
sage: w = s[0]*s[2]
sage: w.first_descent()
0
sage: w = s[0]*s[1]
sage: w.first_descent()
1
has_descent(i, side='right', positive=False)

Returns whether i is a (left/right) descent of self.

See descents() for a description of the options.

EXAMPLES:

sage: W = CoxeterGroups().example()
sage: s = W.simple_reflections()
sage: w = s[0] * s[1] * s[2]
sage: w.has_descent(2)
True
sage: [ w.has_descent(i)                  for i in [0,1,2] ]
[False, False, True]
sage: [ w.has_descent(i, side = 'left')   for i in [0,1,2] ]
[True, False, False]
sage: [ w.has_descent(i, positive = True) for i in [0,1,2] ]
[True, True, False]

This default implementation delegates the work to has_left_descent() and has_right_descent().

has_left_descent(i)

Returns whether \(i\) is a left descent of self.

This default implementation uses that a left descent of \(w\) is a right descent of \(w^{-1}\).

EXAMPLES:

sage: W = CoxeterGroups().example(); W
The symmetric group on {0, ..., 3}
sage: w = W.an_element(); w
(1, 2, 3, 0)
sage: w.has_left_descent(0)
True
sage: w.has_left_descent(1)
False
sage: w.has_left_descent(2)
False

TESTS:

sage: w.has_left_descent.__module__
'sage.categories.coxeter_groups'
has_right_descent(i)

Returns whether i is a right descent of self.

EXAMPLES:

sage: W = CoxeterGroups().example(); W
The symmetric group on {0, ..., 3}
sage: w = W.an_element(); w
(1, 2, 3, 0)
sage: w.has_right_descent(0)
False
sage: w.has_right_descent(1)
False
sage: w.has_right_descent(2)
True
inverse()

Returns the inverse of self

EXAMPLES:

sage: W=WeylGroup(['B',7])
sage: w=W.an_element()
sage: u=w.inverse()
sage: u==~w
True
sage: u*w==w*u
True
sage: u*w
[1 0 0 0 0 0 0]
[0 1 0 0 0 0 0]
[0 0 1 0 0 0 0]
[0 0 0 1 0 0 0]
[0 0 0 0 1 0 0]
[0 0 0 0 0 1 0]
[0 0 0 0 0 0 1]
inversions_as_reflections()

Returns the set of reflections r such that self r < self.

EXAMPLES:

sage: W = WeylGroup(['A',3], prefix="s")
sage: w = W.from_reduced_word([3,1,2,1])
sage: w.inversions_as_reflections()
[s1, s1*s2*s1, s2, s1*s2*s3*s2*s1]
is_grassmannian(side='right')

INPUT:

  • side - “left” or “right” (default: “right”)

Tests whether self is Grassmannian, i.e. it has at most one descent on the right (resp. on the left).

v EXAMPLES:

sage: W = CoxeterGroups().example(); W
The symmetric group on {0, ..., 3}
sage: s = W.simple_reflections()
sage: W.one().is_grassmannian()
True
sage: s[1].is_grassmannian()
True
sage: (s[1]*s[2]).is_grassmannian()
True
sage: (s[0]*s[1]).is_grassmannian()
True
sage: (s[1]*s[2]*s[1]).is_grassmannian()
False

sage: (s[0]*s[2]*s[1]).is_grassmannian(side = "left")
False
sage: (s[0]*s[2]*s[1]).is_grassmannian(side = "right")
True
sage: (s[0]*s[2]*s[1]).is_grassmannian()
True
left_inversions_as_reflections()

Returns the set of reflections r such that r self < self.

EXAMPLES:

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

Returns the length of self, that is the minimal length of a product of simple reflections giving self.

EXAMPLES:

sage: W = CoxeterGroups().example()
sage: s1 = W.simple_reflection(1)
sage: s2 = W.simple_reflection(2)
sage: s1.length()
1
sage: (s1*s2).length()
2
sage: W = CoxeterGroups().example()
sage: s = W.simple_reflections()
sage: w = s[0]*s[1]*s[0]
sage: w.length()
3
sage: W = CoxeterGroups().example()
sage: sum((x^w.length()) for w in W) - expand(prod(sum(x^i for i in range(j+1)) for j in range(4))) # This is scandalously slow!!!
0

SEE ALSO: reduced_word()

TODO: Should use reduced_word_iterator (or reverse_iterator)

lower_cover_reflections(side='right')

Returns the reflections t such that self covers self t.

If side is ‘left’, self covers t self.

EXAMPLES:

sage: W = WeylGroup(['A',3],prefix="s")
sage: w = W.from_reduced_word([3,1,2,1])
sage: w.lower_cover_reflections()
[s1*s2*s3*s2*s1, s2, s1]
sage: w.lower_cover_reflections(side = 'left')
[s2*s3*s2, s3, s1]
lower_covers(side='right', index_set=None)

Returns all elements that self covers in weak order.

INPUT:

  • side - ‘left’ or ‘right’ (default: ‘right’)
  • index_set - a list of indices or None

OUTPUT: a list

EXAMPLES:

sage: W = WeylGroup(['A',3])
sage: w = W.from_reduced_word([3,2,1])
sage: [x.reduced_word() for x in w.lower_covers()]
[[3, 2]]

To obtain covers for left weak order, set the option side to ‘left’:

sage: [x.reduced_word() for x in w.lower_covers(side='left')]
[[2, 1]]
sage: w = W.from_reduced_word([3,2,3,1])
sage: [x.reduced_word() for x in w.lower_covers()]
[[2, 3, 2], [3, 2, 1]]

Covers w.r.t. a parabolic subgroup are obtained with the option index_set:

sage: [x.reduced_word() for x in w.lower_covers(index_set = [1,2])]
[[2, 3, 2]]
sage: [x.reduced_word() for x in w.lower_covers(side='left')]
[[3, 2, 1], [2, 3, 1]]
min_demazure_product_greater(element)

Finds the unique Bruhat-minimum element u such that v \(\le\) w * u where v is self, w is element and * is the Demazure product.

INPUT:

  • element is either an element of the same Coxeter group as self or a list (such as a reduced word) of elements from the index set of the Coxeter group.

EXAMPLES:

sage: W = WeylGroup(['A',4],prefix="s")
sage: v = W.from_reduced_word([2,3,4,1,2])
sage: u = W.from_reduced_word([2,3,2,1])
sage: v.min_demazure_product_greater(u)
s4*s2
sage: v.min_demazure_product_greater([2,3,2,1])
s4*s2
sage: v.min_demazure_product_greater((2,3,2,1))
s4*s2
reduced_word()

Returns a reduced word for self.

This is a word \([i_1,i_2,\ldots,i_k]\) of minimal length such that \(s_{i_1} s_{i_2} \cdots s_{i_k}=self\), where \(s\) are the simple reflections.

EXAMPLES:

sage: W=CoxeterGroups().example()
sage: s=W.simple_reflections()
sage: w=s[0]*s[1]*s[2]
sage: w.reduced_word()
[0, 1, 2]
sage: w=s[0]*s[2]
sage: w.reduced_word()
[2, 0]

SEE ALSO: reduced_words(), reduced_word_reverse_iterator(), length()

reduced_word_reverse_iterator()

Returns a reverse iterator on a reduced word for self.

EXAMPLES:

sage: W=CoxeterGroups().example()
sage: s = W.simple_reflections()
sage: sigma = s[0]*s[1]*s[2]
sage: rI=sigma.reduced_word_reverse_iterator()
sage: [i for i in rI]
[2, 1, 0]
sage: s[0]*s[1]*s[2]==sigma
True
sage: sigma.length()
3

SEE ALSO reduced_word()

Default implementation: recursively remove the first right descent until the identity is reached (see first_descent() and apply_simple_reflection()).

reduced_words()

Returns all reduced words for self.

EXAMPLES:

sage: W=CoxeterGroups().example()
sage: s=W.simple_reflections()
sage: w=s[0]*s[2]
sage: w.reduced_words()
[[2, 0], [0, 2]]
sage: W=WeylGroup(['E',6])
sage: w=W.from_reduced_word([2,3,4,2])
sage: w.reduced_words()
[[3, 2, 4, 2], [2, 3, 4, 2], [3, 4, 2, 4]]

TODO: the result should be full featured finite enumerated set (e.g. counting can be done much faster than iterating).

upper_covers(side='right', index_set=None)

Returns all elements that cover self in weak order.

INPUT:

  • side - ‘left’ or ‘right’ (default: ‘right’)
  • index_set - a list of indices or None

OUTPUT: a list

EXAMPLES:

sage: W = WeylGroup(['A',3])
sage: w = W.from_reduced_word([2,3])
sage: [x.reduced_word() for x in w.upper_covers()]
[[2, 3, 1], [2, 3, 2]]

To obtain covers for left weak order, set the option side to ‘left’:

sage: [x.reduced_word() for x in w.upper_covers(side = 'left')]
[[1, 2, 3], [2, 3, 2]]

Covers w.r.t. a parabolic subgroup are obtained with the option index_set:

sage: [x.reduced_word() for x in w.upper_covers(index_set = [1])]
[[2, 3, 1]]
sage: [x.reduced_word() for x in w.upper_covers(side = 'left', index_set = [1])]
[[1, 2, 3]]
weak_covers(side='right', index_set=None, positive=False)

Returns all elements that self covers in weak order.

INPUT:

  • side - ‘left’ or ‘right’ (default: ‘right’)
  • positive - a boolean (default: False)
  • index_set - a list of indices or None

OUTPUT: a list

EXAMPLES:

sage: W = WeylGroup(['A',3])
sage: w = W.from_reduced_word([3,2,1])
sage: [x.reduced_word() for x in w.weak_covers()]
[[3, 2]]

To obtain instead elements that cover self, set positive = True:

sage: [x.reduced_word() for x in w.weak_covers(positive = True)]
[[3, 1, 2, 1], [2, 3, 2, 1]]

To obtain covers for left weak order, set the option side to ‘left’:

sage: [x.reduced_word() for x in w.weak_covers(side='left')]
[[2, 1]]
sage: w = W.from_reduced_word([3,2,3,1])
sage: [x.reduced_word() for x in w.weak_covers()]
[[2, 3, 2], [3, 2, 1]]
sage: [x.reduced_word() for x in w.weak_covers(side='left')]
[[3, 2, 1], [2, 3, 1]]

Covers w.r.t. a parabolic subgroup are obtained with the option index_set:

sage: [x.reduced_word() for x in w.weak_covers(index_set = [1,2])]
[[2, 3, 2]]
weak_le(other, side='right')

comparison in weak order

INPUT:

  • other - an element of the same Coxeter group
  • side - ‘left’ or ‘right’ (default: ‘right’)

OUTPUT: a boolean

Returns whether self <= other in left (resp. right) weak order, that is if ‘v’ can be obtained from ‘v’ by length increasing multiplication by simple reflections on the left (resp. right).

EXAMPLES:

sage: W = WeylGroup(["A",3])
sage: u = W.from_reduced_word([1,2])
sage: v = W.from_reduced_word([1,2,3,2])
sage: u.weak_le(u)
True
sage: u.weak_le(v)
True
sage: v.weak_le(u)
False
sage: v.weak_le(v)
True

Comparison for left weak order is achieved with the option side:

sage: u.weak_le(v, side = 'left')
False

The implementation uses the equivalent condition that any reduced word for \(u\) is a right (resp. left) prefix of some reduced word for \(v\).

Complexity: \(O(l * c)\), where \(l\) is the minimum of the lengths of \(u\) and of \(v\), and \(c\) is the cost of the low level methods first_descent(), has_descent(), apply_simple_reflection(). Those are typically \(O(n)\), where \(n\) is the rank of the Coxeter group.

We now run consistency tests with permutations:

sage: W = WeylGroup(["A",3])
sage: P4 = Permutations(4)
sage: def P4toW(w): return W.from_reduced_word(w.reduced_word())
sage: for u in P4:  # long time (5s on sage.math, 2011)
...       for v in P4:
...           assert u.permutohedron_lequal(v) == P4toW(u).weak_le(P4toW(v))
...           assert u.permutohedron_lequal(v, side='left') == P4toW(u).weak_le(P4toW(v), side='left')
class CoxeterGroups.ParentMethods
bruhat_interval(x, y)

Returns the list of t such that x <= t <= y.

EXAMPLES:

sage: W = WeylGroup("A3", prefix="s")
sage: [s1,s2,s3]=W.simple_reflections()
sage: W.bruhat_interval(s2,s1*s3*s2*s1*s3)
[s1*s2*s3*s2*s1, s2*s3*s2*s1, s3*s1*s2*s1, s1*s2*s3*s1, s1*s2*s3*s2, s3*s2*s1, s2*s3*s1, s2*s3*s2, s1*s2*s1, s3*s1*s2, s1*s2*s3, s2*s1, s3*s2, s2*s3, s1*s2, s2]
sage: W = WeylGroup(['A',2,1], prefix="s")
sage: [s0,s1,s2]=W.simple_reflections()
sage: W.bruhat_interval(1,s0*s1*s2)
[s0*s1*s2, s1*s2, s0*s2, s0*s1, s2, s1, s0, 1]
canonical_representation()

Return the canonical faithful representation of self.

EXAMPLES:

sage: W = WeylGroup("A3")
sage: W.canonical_representation()
Coxeter group over Universal Cyclotomic Field with Coxeter matrix:
[1 3 2]
[3 1 3]
[2 3 1]
demazure_product(Q)

Returns the Demazure product of the list Q in self.

INPUT:

  • Q is a list of elements from the index set of self.

This returns the Coxeter group element that represents the composition of 0-Hecke or Demazure operators. See CoxeterGroups.ParentMethods.simple_projections().

EXAMPLES:

sage: W = WeylGroup(['A',2])
sage: w = W.demazure_product([2,2,1])
sage: w.reduced_word()
[2, 1]

sage: w = W.demazure_product([2,1,2,1,2])
sage: w.reduced_word()
[1, 2, 1]

sage: W = WeylGroup(['B',2])
sage: w = W.demazure_product([2,1,2,1,2])
sage: w.reduced_word()
[2, 1, 2, 1]
from_reduced_word(word)

INPUT:

  • word - a list (or iterable) of elements of self.index_set()

Returns the group element corresponding to the given word. Namely, if word is \([i_1,i_2,\ldots,i_k]\), then this returns the corresponding product of simple reflections \(s_{i_1} s_{i_2} \cdots s_{i_k}\).

Note: the main use case is for constructing elements from reduced words, hence the name of this method. But actually the input word need not be reduced.

EXAMPLES:

sage: W = CoxeterGroups().example()
sage: W
The symmetric group on {0, ..., 3}
sage: s = W.simple_reflections()
sage: W.from_reduced_word([0,2,0,1])
(0, 3, 1, 2)
sage: W.from_reduced_word((0,2,0,1))
(0, 3, 1, 2)
sage: s[0]*s[2]*s[0]*s[1]
(0, 3, 1, 2)

See also :meth:’._test_reduced_word’:

sage: W._test_reduced_word()

TESTS:

sage: W=WeylGroup(['E',6])
sage: W.from_reduced_word([2,3,4,2])
[ 0  1  0  0  0  0  0  0]
[ 0  0 -1  0  0  0  0  0]
[-1  0  0  0  0  0  0  0]
[ 0  0  0  1  0  0  0  0]
[ 0  0  0  0  1  0  0  0]
[ 0  0  0  0  0  1  0  0]
[ 0  0  0  0  0  0  1  0]
[ 0  0  0  0  0  0  0  1]
grassmannian_elements(side='right')

INPUT:

  • side: “left” or “right” (default: “right”)

Returns the left or right grassmanian elements of self, as an enumerated set

EXAMPLES:

sage: S = CoxeterGroups().example()
sage: G = S.grassmannian_elements()
sage: G.cardinality()
12
sage: G.list()
[(0, 1, 2, 3), (1, 0, 2, 3), (2, 0, 1, 3), (3, 0, 1, 2), (0, 2, 1, 3), (1, 2, 0, 3), (0, 3, 1, 2), (1, 3, 0, 2), (2, 3, 0, 1), (0, 1, 3, 2), (0, 2, 3, 1), (1, 2, 3, 0)]
sage: sorted(tuple(w.descents()) for w in G)
[(), (0,), (0,), (0,), (1,), (1,), (1,), (1,), (1,), (2,), (2,), (2,)]
sage: G = S.grassmannian_elements(side = "left")
sage: G.cardinality()
12
sage: sorted(tuple(w.descents(side = "left")) for w in G)
[(), (0,), (0,), (0,), (1,), (1,), (1,), (1,), (1,), (2,), (2,), (2,)]
group_generators()

Implements Groups.ParentMethods.group_generators() by returning the simple reflections of self.

EXAMPLES:

sage: D10 = FiniteCoxeterGroups().example(10)
sage: D10.group_generators()
Finite family {1: (1,), 2: (2,)}
sage: SymmetricGroup(5).group_generators()
Finite family {1: (1,2), 2: (2,3), 3: (3,4), 4: (4,5)}

Those give semigroup generators, even for an infinite group:

sage: W = WeylGroup(["A",2,1])
sage: W.semigroup_generators()
Finite family {0: [-1  1  1]
                  [ 0  1  0]
                  [ 0  0  1],
               1: [ 1  0  0]
                  [ 1 -1  1]
                  [ 0  0  1],
               2: [ 1  0  0]
                  [ 0  1  0]
                  [ 1  1 -1]}
index_set()

Returns the index set of (the simple reflections of) self, as a list (or iterable).

EXAMPLES:

sage: W = FiniteCoxeterGroups().example(); W
The 5-th dihedral group of order 10
sage: W.index_set()
[1, 2]
semigroup_generators()

Implements Groups.ParentMethods.group_generators() by returning the simple reflections of self.

EXAMPLES:

sage: D10 = FiniteCoxeterGroups().example(10)
sage: D10.group_generators()
Finite family {1: (1,), 2: (2,)}
sage: SymmetricGroup(5).group_generators()
Finite family {1: (1,2), 2: (2,3), 3: (3,4), 4: (4,5)}

Those give semigroup generators, even for an infinite group:

sage: W = WeylGroup(["A",2,1])
sage: W.semigroup_generators()
Finite family {0: [-1  1  1]
                  [ 0  1  0]
                  [ 0  0  1],
               1: [ 1  0  0]
                  [ 1 -1  1]
                  [ 0  0  1],
               2: [ 1  0  0]
                  [ 0  1  0]
                  [ 1  1 -1]}
simple_projection(i, side='right', length_increasing=True, toward_max=None)

INPUT:

  • i - an element of the index set of self

Returns the simple projection \(\pi_i\) (or \(\overline\pi_i\) if \(length_increasing\) is False).

See simple_projections() for the options and for the definition of the simple projections.

EXAMPLES:

sage: W = CoxeterGroups().example()
sage: W
The symmetric group on {0, ..., 3}
sage: s = W.simple_reflections()
sage: sigma=W.an_element()
sage: sigma
(1, 2, 3, 0)
sage: u0=W.simple_projection(0)
sage: d0=W.simple_projection(0,length_increasing=False)
sage: sigma.length()
3
sage: pi=sigma*s[0]
sage: pi.length()
4
sage: u0(sigma)
(2, 1, 3, 0)
sage: pi
(2, 1, 3, 0)
sage: u0(pi)
(2, 1, 3, 0)
sage: d0(sigma)
(1, 2, 3, 0)
sage: d0(pi)
(1, 2, 3, 0)
simple_projections(side='right', length_increasing=True, toward_max=None)

Returns the family of simple projections, also known as 0-Hecke or Demazure operators.

INPUT:

  • self - a Coxeter group \(W\)
  • side - ‘left’ or ‘right’ (default: ‘right’)
  • length_increasing - a boolean (default: True) specifying whether the operator increases or decreases length

Returns the simple projections of \(W\), as a family.

To each simple reflection \(s_i\) of \(W\), corresponds a simple projection \(\pi_i\) from \(W\) to \(W\) defined by:

\(\pi_i(w) = w s_i\) if \(i\) is not a descent of \(w\) \(\pi_i(w) = w\) otherwise.

The simple projections \((\pi_i)_{i\in I}\) move elements down the right permutohedron, toward the maximal element. They satisfy the same braid relations as the simple reflections, but are idempotents \(\pi_i^2=\pi\) not involutions \(s_i^2 = 1\). As such, the simple projections generate the \(0\)-Hecke monoid.

By symmetry, one can also define the projections \((\overline\pi_i)_{i\in I}\) (when the option length_increasing is False):

\(\overline\pi_i(w) = w s_i\) if \(i\) is a descent of \(w\) \(\overline\pi_i(w) = w\) otherwise.

as well as the analogues acting on the left (when the option side is ‘left’).

EXAMPLES:

sage: W = CoxeterGroups().example()
sage: W
The symmetric group on {0, ..., 3}
sage: s = W.simple_reflections()
sage: sigma=W.an_element()
sage: sigma
(1, 2, 3, 0)
sage: pi=W.simple_projections()
sage: pi
Finite family {0: <function <lambda> at ...>, 1: <function <lambda> at ...>, 2: <function <lambda> ...>}
sage: pi[1](sigma)
(1, 3, 2, 0)
sage: W.simple_projection(1)(sigma)
(1, 3, 2, 0)
simple_reflection(i)

INPUT:

  • i - an element from the index set.

Returns the simple reflection \(s_i\)

EXAMPLES:

sage: W = CoxeterGroups().example()
sage: W
The symmetric group on {0, ..., 3}
sage: W.simple_reflection(1)
(0, 2, 1, 3)
sage: s = W.simple_reflections()
sage: s[1]
(0, 2, 1, 3)
simple_reflections()

Returns the simple reflections \((s_i)_{i\in I}\), as a family.

EXAMPLES:

sage: W = CoxeterGroups().example()
sage: W
The symmetric group on {0, ..., 3}
sage: s = W.simple_reflections()
sage: s
Finite family {0: (1, 0, 2, 3), 1: (0, 2, 1, 3), 2: (0, 1, 3, 2)}
sage: s[0]
(1, 0, 2, 3)
sage: s[1]
(0, 2, 1, 3)
sage: s[2]
(0, 1, 3, 2)

This default implementation uses index_set() and simple_reflection().

some_elements()

Implements Sets.ParentMethods.some_elements() by returning some typical element of \(self\).

EXAMPLES:

sage: W=WeylGroup(['A',3])
sage: W.some_elements()
[
[0 1 0 0]  [1 0 0 0]  [1 0 0 0]  [1 0 0 0]  [0 0 0 1]
[1 0 0 0]  [0 0 1 0]  [0 1 0 0]  [0 1 0 0]  [1 0 0 0]
[0 0 1 0]  [0 1 0 0]  [0 0 0 1]  [0 0 1 0]  [0 1 0 0]
[0 0 0 1], [0 0 0 1], [0 0 1 0], [0 0 0 1], [0 0 1 0]
]
sage: W.order()
24
weak_order_ideal(predicate, side='right', category=None)

Returns a weak order ideal defined by a predicate

INPUT:

  • predicate: a predicate on the elements of self defining an weak order ideal in self
  • side: “left” or “right” (default: “right”)

OUTPUT: an enumerated set

EXAMPLES:

sage: D6 = FiniteCoxeterGroups().example(5)
sage: I = D6.weak_order_ideal(predicate = lambda w: w.length() <= 3)
sage: I.cardinality()
7
sage: list(I)
[(), (1,), (1, 2), (1, 2, 1), (2,), (2, 1), (2, 1, 2)]

We now consider an infinite Coxeter group:

sage: W = WeylGroup(["A",1,1])
sage: I = W.weak_order_ideal(predicate = lambda w: w.length() <= 2)
sage: list(iter(I))
[
[1 0]  [-1  2]  [ 3 -2]  [ 1  0]  [-1  2]
[0 1], [ 0  1], [ 2 -1], [ 2 -1], [-2  3]
]

Even when the result is finite, some features of FiniteEnumeratedSets are not available:

sage: I.cardinality() # todo: not implemented
5
sage: list(I)         # todo: not implemented

unless this finiteness is explicitly specified:

sage: I = W.weak_order_ideal(predicate = lambda w: w.length() <= 2,
...                          category = FiniteEnumeratedSets())
sage: I.cardinality()
5
sage: list(I)
[
[1 0]  [-1  2]  [ 3 -2]  [ 1  0]  [-1  2]
[0 1], [ 0  1], [ 2 -1], [ 2 -1], [-2  3]
]

Background

The weak order is returned as a SearchForest. This is achieved by assigning to each element \(u1\) of the ideal a single ancestor \(u=u1 s_i\), where \(i\) is the smallest descent of \(u\).

This allows for iterating through the elements in roughly Constant Amortized Time and constant memory (taking the operations and size of the generated objects as constants).

CoxeterGroups.super_categories()

EXAMPLES:

sage: CoxeterGroups().super_categories()
[Category of groups, Category of enumerated sets]

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