Finite Coxeter Groups

class sage.categories.finite_coxeter_groups.FiniteCoxeterGroups(base_category)[source]

Bases: CategoryWithAxiom

The category of finite Coxeter groups.

EXAMPLES:

sage: CoxeterGroups.Finite()
Category of finite Coxeter groups
sage: FiniteCoxeterGroups().super_categories()
[Category of finite generalized Coxeter groups,
 Category of Coxeter groups]

sage: G = CoxeterGroups().Finite().example()
sage: G.cayley_graph(side = "right").plot()
Graphics object consisting of 40 graphics primitives

>>> from sage.all import *
>>> CoxeterGroups.Finite()
Category of finite Coxeter groups
>>> FiniteCoxeterGroups().super_categories()
[Category of finite generalized Coxeter groups,
 Category of Coxeter groups]

>>> G = CoxeterGroups().Finite().example()
>>> G.cayley_graph(side = "right").plot()
Graphics object consisting of 40 graphics primitives

Here are some further examples:

sage: WeylGroups().Finite().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)

>>> from sage.all import *
>>> WeylGroups().Finite().example()
The symmetric group on {0, ..., 3}

>>> WeylGroup(["B", Integer(3)])
Weyl Group of type ['B', 3] (as a matrix group acting on the ambient space)

Those other examples 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

>>> from sage.all import *
>>> SymmetricGroup(Integer(4))
Symmetric group of order 4! as a permutation group
>>> DihedralGroup(Integer(5))
Dihedral group of order 10 as a permutation group
class ElementMethods[source]

Bases: object

absolute_length()[source]

Return the absolute length of self.

The absolute length is the length of the shortest expression of the element as a product of reflections. For finite Coxeter groups, the absolute length is the codimension of the 1-eigenspace of the element (Lemmas 1-3 in [Car1972a]).

For permutations in the symmetric groups, the absolute length is the size minus the number of its disjoint cycles.

See also

absolute_le()

EXAMPLES:

sage: W = WeylGroup(["A", 3])                                           # needs sage.combinat sage.groups
sage: s = W.simple_reflections()                                        # needs sage.combinat sage.groups
sage: (s[1]*s[2]*s[3]).absolute_length()                                # needs sage.combinat sage.groups
3

sage: W = SymmetricGroup(4)                                             # needs sage.groups
sage: s = W.simple_reflections()                                        # needs sage.groups
sage: (s[3]*s[2]*s[1]).absolute_length()                                # needs sage.combinat sage.groups
3

>>> from sage.all import *
>>> W = WeylGroup(["A", Integer(3)])                                           # needs sage.combinat sage.groups
>>> s = W.simple_reflections()                                        # needs sage.combinat sage.groups
>>> (s[Integer(1)]*s[Integer(2)]*s[Integer(3)]).absolute_length()                                # needs sage.combinat sage.groups
3

>>> W = SymmetricGroup(Integer(4))                                             # needs sage.groups
>>> s = W.simple_reflections()                                        # needs sage.groups
>>> (s[Integer(3)]*s[Integer(2)]*s[Integer(1)]).absolute_length()                                # needs sage.combinat sage.groups
3
bruhat_upper_covers()[source]

Return all the elements that cover self in Bruhat order.

EXAMPLES:

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

sage: W = WeylGroup(["B",6])
sage: w = W.from_reduced_word([1,2,1,4,5])
sage: C = w.bruhat_upper_covers()
sage: len(C)
9
sage: print([v.reduced_word() for v in C])
[[6, 4, 5, 1, 2, 1], [4, 5, 6, 1, 2, 1], [3, 4, 5, 1, 2, 1], [2, 3, 4, 5, 1, 2],
[1, 2, 3, 4, 5, 1], [4, 5, 4, 1, 2, 1], [4, 5, 3, 1, 2, 1], [4, 5, 2, 3, 1, 2],
[4, 5, 1, 2, 3, 1]]
sage: ww = W.from_reduced_word([5,6,5])
sage: CC = ww.bruhat_upper_covers()
sage: print([v.reduced_word() for v in CC])
[[6, 5, 6, 5], [4, 5, 6, 5], [5, 6, 4, 5], [5, 6, 5, 4], [5, 6, 5, 3], [5, 6, 5, 2],
[5, 6, 5, 1]]

>>> from sage.all import *
>>> W = WeylGroup(["A",Integer(4)])
>>> w = W.from_reduced_word([Integer(3),Integer(2)])
>>> print([v.reduced_word() for v in w.bruhat_upper_covers()])
[[4, 3, 2], [3, 4, 2], [2, 3, 2], [3, 1, 2], [3, 2, 1]]

>>> W = WeylGroup(["B",Integer(6)])
>>> w = W.from_reduced_word([Integer(1),Integer(2),Integer(1),Integer(4),Integer(5)])
>>> C = w.bruhat_upper_covers()
>>> len(C)
9
>>> print([v.reduced_word() for v in C])
[[6, 4, 5, 1, 2, 1], [4, 5, 6, 1, 2, 1], [3, 4, 5, 1, 2, 1], [2, 3, 4, 5, 1, 2],
[1, 2, 3, 4, 5, 1], [4, 5, 4, 1, 2, 1], [4, 5, 3, 1, 2, 1], [4, 5, 2, 3, 1, 2],
[4, 5, 1, 2, 3, 1]]
>>> ww = W.from_reduced_word([Integer(5),Integer(6),Integer(5)])
>>> CC = ww.bruhat_upper_covers()
>>> print([v.reduced_word() for v in CC])
[[6, 5, 6, 5], [4, 5, 6, 5], [5, 6, 4, 5], [5, 6, 5, 4], [5, 6, 5, 3], [5, 6, 5, 2],
[5, 6, 5, 1]]

Recursive algorithm: write \(w\) for self. If \(i\) is a non-descent of \(w\), then the covers of \(w\) are exactly \(\{ws_i, u_1s_i, u_2s_i,..., u_js_i\}\), where the \(u_k\) are those covers of \(ws_i\) that have a descent at \(i\).

covered_reflections_subgroup()[source]

Return the subgroup of \(W\) generated by the conjugates by \(w\) of the simple reflections indexed by right descents of \(w\).

This is used to compute the shard intersection order on \(W\).

EXAMPLES:

sage: W = CoxeterGroup(['A',3], base_ring=ZZ)
sage: len(W.long_element().covered_reflections_subgroup())
24
sage: s = W.simple_reflection(1)
sage: Gs = s.covered_reflections_subgroup()
sage: len(Gs)
2
sage: s in [u.lift() for u in Gs]
True
sage: len(W.one().covered_reflections_subgroup())
1

>>> from sage.all import *
>>> W = CoxeterGroup(['A',Integer(3)], base_ring=ZZ)
>>> len(W.long_element().covered_reflections_subgroup())
24
>>> s = W.simple_reflection(Integer(1))
>>> Gs = s.covered_reflections_subgroup()
>>> len(Gs)
2
>>> s in [u.lift() for u in Gs]
True
>>> len(W.one().covered_reflections_subgroup())
1
coxeter_knuth_graph()[source]

Return the Coxeter-Knuth graph of type \(A\).

The Coxeter-Knuth graph of type \(A\) is generated by the Coxeter-Knuth relations which are given by \(a a+1 a \sim a+1 a a+1\), \(abc \sim acb\) if \(b<a<c\) and \(abc \sim bac\) if \(a<c<b\).

EXAMPLES:

sage: W = WeylGroup(['A',4], prefix='s')
sage: w = W.from_reduced_word([1,2,1,3,2])
sage: D = w.coxeter_knuth_graph()
sage: D.vertices(sort=True)
[(1, 2, 1, 3, 2),
(1, 2, 3, 1, 2),
(2, 1, 2, 3, 2),
(2, 1, 3, 2, 3),
(2, 3, 1, 2, 3)]
sage: D.edges(sort=True)
[((1, 2, 1, 3, 2), (1, 2, 3, 1, 2), None),
((1, 2, 1, 3, 2), (2, 1, 2, 3, 2), None),
((2, 1, 2, 3, 2), (2, 1, 3, 2, 3), None),
((2, 1, 3, 2, 3), (2, 3, 1, 2, 3), None)]

sage: w = W.from_reduced_word([1,3])
sage: D = w.coxeter_knuth_graph()
sage: D.vertices(sort=True)
[(1, 3), (3, 1)]
sage: D.edges(sort=False)
[]

>>> from sage.all import *
>>> W = WeylGroup(['A',Integer(4)], prefix='s')
>>> w = W.from_reduced_word([Integer(1),Integer(2),Integer(1),Integer(3),Integer(2)])
>>> D = w.coxeter_knuth_graph()
>>> D.vertices(sort=True)
[(1, 2, 1, 3, 2),
(1, 2, 3, 1, 2),
(2, 1, 2, 3, 2),
(2, 1, 3, 2, 3),
(2, 3, 1, 2, 3)]
>>> D.edges(sort=True)
[((1, 2, 1, 3, 2), (1, 2, 3, 1, 2), None),
((1, 2, 1, 3, 2), (2, 1, 2, 3, 2), None),
((2, 1, 2, 3, 2), (2, 1, 3, 2, 3), None),
((2, 1, 3, 2, 3), (2, 3, 1, 2, 3), None)]

>>> w = W.from_reduced_word([Integer(1),Integer(3)])
>>> D = w.coxeter_knuth_graph()
>>> D.vertices(sort=True)
[(1, 3), (3, 1)]
>>> D.edges(sort=False)
[]
coxeter_knuth_neighbor(w)[source]

Return the Coxeter-Knuth (oriented) neighbors of the reduced word \(w\) of self.

INPUT:

  • w – reduced word of self

The Coxeter-Knuth relations are given by \(a a+1 a \sim a+1 a a+1\), \(abc \sim acb\) if \(b<a<c\) and \(abc \sim bac\) if \(a<c<b\). This method returns all neighbors of w under the Coxeter-Knuth relations oriented from left to right.

EXAMPLES:

sage: W = WeylGroup(['A',4], prefix='s')
sage: word = [1,2,1,3,2]
sage: w = W.from_reduced_word(word)
sage: w.coxeter_knuth_neighbor(word)
{(1, 2, 3, 1, 2), (2, 1, 2, 3, 2)}

sage: word = [1,2,1,3,2,4,3]
sage: w = W.from_reduced_word(word)
sage: w.coxeter_knuth_neighbor(word)
{(1, 2, 1, 3, 4, 2, 3), (1, 2, 3, 1, 2, 4, 3), (2, 1, 2, 3, 2, 4, 3)}

>>> from sage.all import *
>>> W = WeylGroup(['A',Integer(4)], prefix='s')
>>> word = [Integer(1),Integer(2),Integer(1),Integer(3),Integer(2)]
>>> w = W.from_reduced_word(word)
>>> w.coxeter_knuth_neighbor(word)
{(1, 2, 3, 1, 2), (2, 1, 2, 3, 2)}

>>> word = [Integer(1),Integer(2),Integer(1),Integer(3),Integer(2),Integer(4),Integer(3)]
>>> w = W.from_reduced_word(word)
>>> w.coxeter_knuth_neighbor(word)
{(1, 2, 1, 3, 4, 2, 3), (1, 2, 3, 1, 2, 4, 3), (2, 1, 2, 3, 2, 4, 3)}
is_coxeter_element()[source]

Return whether this is a Coxeter element.

This is, whether self has an eigenvalue \(e^{2\pi i/h}\) where \(h\) is the Coxeter number.

See also

coxeter_elements()

EXAMPLES:

sage: W = CoxeterGroup(['A',2])
sage: c = prod(W.gens())
sage: c.is_coxeter_element()
True
sage: W.one().is_coxeter_element()
False

sage: W = WeylGroup(['G', 2])
sage: c = prod(W.gens())
sage: c.is_coxeter_element()
True
sage: W.one().is_coxeter_element()
False

>>> from sage.all import *
>>> W = CoxeterGroup(['A',Integer(2)])
>>> c = prod(W.gens())
>>> c.is_coxeter_element()
True
>>> W.one().is_coxeter_element()
False

>>> W = WeylGroup(['G', Integer(2)])
>>> c = prod(W.gens())
>>> c.is_coxeter_element()
True
>>> W.one().is_coxeter_element()
False
class ParentMethods[source]

Bases: object

Ambiguity resolution: the implementation of some_elements is preferable to that of FiniteGroups. The same holds for __iter__, although a breadth first search would be more natural; at least this maintains backward compatibility after Issue #13589.

bhz_poset()[source]

Return the Bergeron-Hohlweg-Zabrocki partial order on the Coxeter group.

This is a partial order on the elements of a finite Coxeter group \(W\), which is distinct from the Bruhat order, the weak order and the shard intersection order. It was defined in [BHZ2005].

This partial order is not a lattice, as there is no unique maximal element. It can be succinctly defined as follows.

Let \(u\) and \(v\) be two elements of the Coxeter group \(W\). Let \(S(u)\) be the support of \(u\). Then \(u \leq v\) if and only if \(v_{S(u)} = u\) (here \(v = v^I v_I\) denotes the usual parabolic decomposition with respect to the standard parabolic subgroup \(W_I\)).

See also

bruhat_poset(), shard_poset(), weak_poset()

EXAMPLES:

sage: W = CoxeterGroup(['A',3], base_ring=ZZ)
sage: P = W.bhz_poset(); P
Finite poset containing 24 elements
sage: P.relations_number()
103
sage: P.chain_polynomial()
34*q^4 + 90*q^3 + 79*q^2 + 24*q + 1
sage: len(P.maximal_elements())
13

>>> from sage.all import *
>>> W = CoxeterGroup(['A',Integer(3)], base_ring=ZZ)
>>> P = W.bhz_poset(); P
Finite poset containing 24 elements
>>> P.relations_number()
103
>>> P.chain_polynomial()
34*q^4 + 90*q^3 + 79*q^2 + 24*q + 1
>>> len(P.maximal_elements())
13
bruhat_poset(facade=False)[source]

Return the Bruhat poset of self.

See also

bhz_poset(), shard_poset(), weak_poset()

EXAMPLES:

sage: W = WeylGroup(["A", 2])
sage: P = W.bruhat_poset()
sage: P
Finite poset containing 6 elements
sage: P.show()

>>> from sage.all import *
>>> W = WeylGroup(["A", Integer(2)])
>>> P = W.bruhat_poset()
>>> P
Finite poset containing 6 elements
>>> P.show()

Here are some typical operations on this poset:

sage: W = WeylGroup(["A", 3])
sage: P = W.bruhat_poset()
sage: u = W.from_reduced_word([3,1])
sage: v = W.from_reduced_word([3,2,1,2,3])
sage: P(u) <= P(v)
True
sage: len(P.interval(P(u), P(v)))
10
sage: P.is_join_semilattice()
False

>>> from sage.all import *
>>> W = WeylGroup(["A", Integer(3)])
>>> P = W.bruhat_poset()
>>> u = W.from_reduced_word([Integer(3),Integer(1)])
>>> v = W.from_reduced_word([Integer(3),Integer(2),Integer(1),Integer(2),Integer(3)])
>>> P(u) <= P(v)
True
>>> len(P.interval(P(u), P(v)))
10
>>> P.is_join_semilattice()
False

By default, the elements of \(P\) are aware that they belong to \(P\):

sage: P.an_element().parent()
Finite poset containing 24 elements

>>> from sage.all import *
>>> P.an_element().parent()
Finite poset containing 24 elements

If instead one wants the elements to be plain elements of the Coxeter group, one can use the facade option:

sage: P = W.bruhat_poset(facade = True)
sage: P.an_element().parent()
Weyl Group of type ['A', 3] (as a matrix group acting on the ambient space)

>>> from sage.all import *
>>> P = W.bruhat_poset(facade = True)
>>> P.an_element().parent()
Weyl Group of type ['A', 3] (as a matrix group acting on the ambient space)

See also

Poset() for more on posets and facade posets.

Todo

  • Use the symmetric group in the examples (for nicer output), and print the edges for a stronger test.

  • The constructed poset should be lazy, in order to handle large / infinite Coxeter groups.

cambrian_lattice(c, on_roots=False)[source]

Return the \(c\)-Cambrian lattice on delta sequences.

See arXiv 1503.00710 and arXiv math/0611106.

Delta sequences are certain 2-colored minimal factorizations of \(c\) into reflections.

INPUT:

  • c – a standard Coxeter element in self (as a tuple, or as an element of self)

  • on_roots – boolean (default: False); if on_roots is True, the lattice is realized on roots rather than on reflections. In order for this to work, the ElementMethod reflection_to_root must be available.

EXAMPLES:

sage: CoxeterGroup(["A", 2]).cambrian_lattice((1,2))
Finite lattice containing 5 elements

sage: CoxeterGroup(["B", 2]).cambrian_lattice((1,2))
Finite lattice containing 6 elements

sage: CoxeterGroup(["G", 2]).cambrian_lattice((1,2))
Finite lattice containing 8 elements

>>> from sage.all import *
>>> CoxeterGroup(["A", Integer(2)]).cambrian_lattice((Integer(1),Integer(2)))
Finite lattice containing 5 elements

>>> CoxeterGroup(["B", Integer(2)]).cambrian_lattice((Integer(1),Integer(2)))
Finite lattice containing 6 elements

>>> CoxeterGroup(["G", Integer(2)]).cambrian_lattice((Integer(1),Integer(2)))
Finite lattice containing 8 elements
codegrees()[source]

Return the codegrees of the Coxeter group.

These are just the degrees minus 2.

EXAMPLES:

sage: CoxeterGroup(['A', 4]).codegrees()
(0, 1, 2, 3)
sage: CoxeterGroup(['B', 4]).codegrees()
(0, 2, 4, 6)
sage: CoxeterGroup(['D', 4]).codegrees()
(0, 2, 2, 4)
sage: CoxeterGroup(['F', 4]).codegrees()
(0, 4, 6, 10)
sage: CoxeterGroup(['E', 8]).codegrees()
(0, 6, 10, 12, 16, 18, 22, 28)
sage: CoxeterGroup(['H', 3]).codegrees()
(0, 4, 8)

sage: WeylGroup([["A",3], ["A",3], ["B",2]]).codegrees()
(0, 1, 2, 0, 1, 2, 0, 2)

>>> from sage.all import *
>>> CoxeterGroup(['A', Integer(4)]).codegrees()
(0, 1, 2, 3)
>>> CoxeterGroup(['B', Integer(4)]).codegrees()
(0, 2, 4, 6)
>>> CoxeterGroup(['D', Integer(4)]).codegrees()
(0, 2, 2, 4)
>>> CoxeterGroup(['F', Integer(4)]).codegrees()
(0, 4, 6, 10)
>>> CoxeterGroup(['E', Integer(8)]).codegrees()
(0, 6, 10, 12, 16, 18, 22, 28)
>>> CoxeterGroup(['H', Integer(3)]).codegrees()
(0, 4, 8)

>>> WeylGroup([["A",Integer(3)], ["A",Integer(3)], ["B",Integer(2)]]).codegrees()
(0, 1, 2, 0, 1, 2, 0, 2)
coxeter_complex()[source]

Return the Coxeter complex of self.

Let \(W\) be a Coxeter group, and let \(X\) be the corresponding Tits cone, which is constructed as the \(W\) orbit of the fundamental chamber in the reflection representation. The Coxeter complex of \(W\) is the simplicial complex \((X \setminus \{0\}) / \RR_{>0}\). The face poset of this simplicial complex is given by the coxeter_poset(). When \(W\) is a finite group, then the Coxeter complex is homeomorphic to an \((n-1)\)-dimensional sphere, where \(n\) is the rank of \(W\).

EXAMPLES:

sage: W = CoxeterGroup(['A', 3])
sage: C = W.coxeter_complex()
sage: C
Simplicial complex with 14 vertices and 24 facets
sage: C.homology()
{0: 0, 1: 0, 2: Z}

sage: W = WeylGroup(['B', 3])
sage: C = W.coxeter_complex()
sage: C
Simplicial complex with 26 vertices and 48 facets
sage: C.homology()
{0: 0, 1: 0, 2: Z}

sage: W = CoxeterGroup(['I', 7])
sage: C = W.coxeter_complex()
sage: C
Simplicial complex with 14 vertices and 14 facets
sage: C.homology()
{0: 0, 1: Z}

sage: W = CoxeterGroup(['H', 3])
sage: C = W.coxeter_complex()
sage: C
Simplicial complex with 62 vertices and 120 facets
sage: C.homology()
{0: 0, 1: 0, 2: Z}

sage: # optional - gap3
sage: W = CoxeterGroup(['H', 3], implementation='permutation')
sage: C = W.coxeter_complex()
sage: C
Simplicial complex with 62 vertices and 120 facets
sage: C.homology()
{0: 0, 1: 0, 2: Z}

>>> from sage.all import *
>>> W = CoxeterGroup(['A', Integer(3)])
>>> C = W.coxeter_complex()
>>> C
Simplicial complex with 14 vertices and 24 facets
>>> C.homology()
{0: 0, 1: 0, 2: Z}

>>> W = WeylGroup(['B', Integer(3)])
>>> C = W.coxeter_complex()
>>> C
Simplicial complex with 26 vertices and 48 facets
>>> C.homology()
{0: 0, 1: 0, 2: Z}

>>> W = CoxeterGroup(['I', Integer(7)])
>>> C = W.coxeter_complex()
>>> C
Simplicial complex with 14 vertices and 14 facets
>>> C.homology()
{0: 0, 1: Z}

>>> W = CoxeterGroup(['H', Integer(3)])
>>> C = W.coxeter_complex()
>>> C
Simplicial complex with 62 vertices and 120 facets
>>> C.homology()
{0: 0, 1: 0, 2: Z}

>>> # optional - gap3
>>> W = CoxeterGroup(['H', Integer(3)], implementation='permutation')
>>> C = W.coxeter_complex()
>>> C
Simplicial complex with 62 vertices and 120 facets
>>> C.homology()
{0: 0, 1: 0, 2: Z}
coxeter_poset()[source]

Return the Coxeter poset of self.

Let \(W\) be a Coxeter group. The Coxeter poset is defined as the set of (right) standard cosets \(gW_J\), where \(J\) is a subset of the index set \(I\) of \(W\), ordered by reverse inclusion.

This is equal to the face poset of the Coxeter complex.

EXAMPLES:

sage: W = CoxeterGroup(['A', 3])
sage: P = W.coxeter_poset()
sage: P
Finite meet-semilattice containing 75 elements
sage: P.rank()
3

sage: W = WeylGroup(['B', 3])
sage: P = W.coxeter_poset()
sage: P
Finite meet-semilattice containing 147 elements
sage: P.rank()
3

sage: W = CoxeterGroup(['I', 7])
sage: P = W.coxeter_poset()
sage: P
Finite meet-semilattice containing 29 elements
sage: P.rank()
2

sage: W = CoxeterGroup(['H', 3])
sage: P = W.coxeter_poset()
sage: P
Finite meet-semilattice containing 363 elements
sage: P.rank()
3

sage: # optional - gap3
sage: W = CoxeterGroup(['H', 3], implementation='permutation')
sage: P = W.coxeter_poset()
sage: P
Finite meet-semilattice containing 363 elements
sage: P.rank()
3

>>> from sage.all import *
>>> W = CoxeterGroup(['A', Integer(3)])
>>> P = W.coxeter_poset()
>>> P
Finite meet-semilattice containing 75 elements
>>> P.rank()
3

>>> W = WeylGroup(['B', Integer(3)])
>>> P = W.coxeter_poset()
>>> P
Finite meet-semilattice containing 147 elements
>>> P.rank()
3

>>> W = CoxeterGroup(['I', Integer(7)])
>>> P = W.coxeter_poset()
>>> P
Finite meet-semilattice containing 29 elements
>>> P.rank()
2

>>> W = CoxeterGroup(['H', Integer(3)])
>>> P = W.coxeter_poset()
>>> P
Finite meet-semilattice containing 363 elements
>>> P.rank()
3

>>> # optional - gap3
>>> W = CoxeterGroup(['H', Integer(3)], implementation='permutation')
>>> P = W.coxeter_poset()
>>> P
Finite meet-semilattice containing 363 elements
>>> P.rank()
3
degrees()[source]

Return the degrees of the Coxeter group.

The output is an increasing list of integers.

EXAMPLES:

sage: CoxeterGroup(['A', 4]).degrees()
(2, 3, 4, 5)
sage: CoxeterGroup(['B', 4]).degrees()
(2, 4, 6, 8)
sage: CoxeterGroup(['D', 4]).degrees()
(2, 4, 4, 6)
sage: CoxeterGroup(['F', 4]).degrees()
(2, 6, 8, 12)
sage: CoxeterGroup(['E', 8]).degrees()
(2, 8, 12, 14, 18, 20, 24, 30)
sage: CoxeterGroup(['H', 3]).degrees()
(2, 6, 10)

sage: WeylGroup([["A",3], ["A",3], ["B",2]]).degrees()
(2, 3, 4, 2, 3, 4, 2, 4)

>>> from sage.all import *
>>> CoxeterGroup(['A', Integer(4)]).degrees()
(2, 3, 4, 5)
>>> CoxeterGroup(['B', Integer(4)]).degrees()
(2, 4, 6, 8)
>>> CoxeterGroup(['D', Integer(4)]).degrees()
(2, 4, 4, 6)
>>> CoxeterGroup(['F', Integer(4)]).degrees()
(2, 6, 8, 12)
>>> CoxeterGroup(['E', Integer(8)]).degrees()
(2, 8, 12, 14, 18, 20, 24, 30)
>>> CoxeterGroup(['H', Integer(3)]).degrees()
(2, 6, 10)

>>> WeylGroup([["A",Integer(3)], ["A",Integer(3)], ["B",Integer(2)]]).degrees()
(2, 3, 4, 2, 3, 4, 2, 4)
inversion_sequence(word)[source]

Return the inversion sequence corresponding to the word in indices of simple generators of self.

If word corresponds to \([w_0,w_1,...w_k]\), the output is \([w_0,w_0w_1w_0,\ldots,w_0w_1\cdots w_k \cdots w_1 w_0]\).

INPUT:

  • word – a word in the indices of the simple generators of self

EXAMPLES:

sage: CoxeterGroup(["A", 2]).inversion_sequence([1,2,1])
[
[-1  1]  [ 0 -1]  [ 1  0]
[ 0  1], [-1  0], [ 1 -1]
]

sage: [t.reduced_word() for t in CoxeterGroup(["A",3]).inversion_sequence([2,1,3,2,1,3])]
[[2], [1, 2, 1], [2, 3, 2], [1, 2, 3, 2, 1], [3], [1]]

>>> from sage.all import *
>>> CoxeterGroup(["A", Integer(2)]).inversion_sequence([Integer(1),Integer(2),Integer(1)])
[
[-1  1]  [ 0 -1]  [ 1  0]
[ 0  1], [-1  0], [ 1 -1]
]

>>> [t.reduced_word() for t in CoxeterGroup(["A",Integer(3)]).inversion_sequence([Integer(2),Integer(1),Integer(3),Integer(2),Integer(1),Integer(3)])]
[[2], [1, 2, 1], [2, 3, 2], [1, 2, 3, 2, 1], [3], [1]]
is_real()[source]

Return True since self is a real reflection group.

EXAMPLES:

sage: CoxeterGroup(['F',4]).is_real()
True
sage: CoxeterGroup(['H',4]).is_real()
True

>>> from sage.all import *
>>> CoxeterGroup(['F',Integer(4)]).is_real()
True
>>> CoxeterGroup(['H',Integer(4)]).is_real()
True
long_element(index_set=None, as_word=False)[source]

Return the longest element of self, or of the parabolic subgroup corresponding to the given index_set.

INPUT:

  • index_set – a subset (as a list or iterable) of the nodes of the Dynkin diagram; (default: all of them)

  • as_word – boolean (default: False); if True, then return instead a reduced decomposition of the longest element

Should this method be called maximal_element? longest_element?

EXAMPLES:

sage: D10 = FiniteCoxeterGroups().example(10)
sage: D10.long_element()
(1, 2, 1, 2, 1, 2, 1, 2, 1, 2)
sage: D10.long_element([1])
(1,)
sage: D10.long_element([2])
(2,)
sage: D10.long_element([])
()

sage: D7 = FiniteCoxeterGroups().example(7)
sage: D7.long_element()
(1, 2, 1, 2, 1, 2, 1)

>>> from sage.all import *
>>> D10 = FiniteCoxeterGroups().example(Integer(10))
>>> D10.long_element()
(1, 2, 1, 2, 1, 2, 1, 2, 1, 2)
>>> D10.long_element([Integer(1)])
(1,)
>>> D10.long_element([Integer(2)])
(2,)
>>> D10.long_element([])
()

>>> D7 = FiniteCoxeterGroups().example(Integer(7))
>>> D7.long_element()
(1, 2, 1, 2, 1, 2, 1)

One can require instead a reduced word for w0:

sage: A3 = CoxeterGroup(['A', 3])
sage: A3.long_element(as_word=True)
[1, 2, 1, 3, 2, 1]

>>> from sage.all import *
>>> A3 = CoxeterGroup(['A', Integer(3)])
>>> A3.long_element(as_word=True)
[1, 2, 1, 3, 2, 1]
m_cambrian_lattice(c, m=1, on_roots=False)[source]

Return the \(m\)-Cambrian lattice on \(m\)-delta sequences.

See arXiv 1503.00710 and arXiv math/0611106.

The \(m\)-delta sequences are certain \(m\)-colored minimal factorizations of \(c\) into reflections.

INPUT:

  • c – a Coxeter element of self (as a tuple, or as an element of self)

  • m – positive integer (default: 1)

  • on_roots – boolean (default: False); if on_roots is True, the lattice is realized on roots rather than on reflections. In order for this to work, the ElementMethod reflection_to_root must be available.

EXAMPLES:

sage: CoxeterGroup(["A",2]).m_cambrian_lattice((1,2))
Finite lattice containing 5 elements

sage: CoxeterGroup(["A",2]).m_cambrian_lattice((1,2),2)
Finite lattice containing 12 elements

>>> from sage.all import *
>>> CoxeterGroup(["A",Integer(2)]).m_cambrian_lattice((Integer(1),Integer(2)))
Finite lattice containing 5 elements

>>> CoxeterGroup(["A",Integer(2)]).m_cambrian_lattice((Integer(1),Integer(2)),Integer(2))
Finite lattice containing 12 elements
permutahedron(point=None, base_ring=None)[source]

Return the permutahedron of self.

This is the convex hull of the point point in the weight basis under the action of self on the underlying vector space \(V\).

See also

permutahedron()

INPUT:

  • point – (optional) a point given by its coordinates in the weight basis (default: \((1, 1, 1, \ldots)\))

  • base_ring – (optional) the base ring of the polytope

Note

The result is expressed in the root basis coordinates.

Note

If function is too slow, switching the base ring to RDF will almost certainly speed things up.

EXAMPLES:

sage: W = CoxeterGroup(['H',3], base_ring=RDF)
sage: W.permutahedron()
doctest:warning
...
UserWarning: This polyhedron data is numerically complicated; cdd could not convert between the inexact V and H representation without loss of data. The resulting object might show inconsistencies.
A 3-dimensional polyhedron in RDF^3 defined as the convex hull of 120 vertices

sage: W = CoxeterGroup(['I',7])
sage: W.permutahedron()
A 2-dimensional polyhedron in AA^2 defined as the convex hull of 14 vertices
sage: W.permutahedron(base_ring=RDF)
A 2-dimensional polyhedron in RDF^2 defined as the convex hull of 14 vertices

sage: W = ReflectionGroup(['A',3])                          # optional - gap3
sage: W.permutahedron()                                     # optional - gap3
A 3-dimensional polyhedron in QQ^3 defined as the convex hull
of 24 vertices

sage: W = ReflectionGroup(['A',3],['B',2])                  # optional - gap3
sage: W.permutahedron()                                     # optional - gap3
A 5-dimensional polyhedron in QQ^5 defined as the convex hull of 192 vertices

>>> from sage.all import *
>>> W = CoxeterGroup(['H',Integer(3)], base_ring=RDF)
>>> W.permutahedron()
doctest:warning
...
UserWarning: This polyhedron data is numerically complicated; cdd could not convert between the inexact V and H representation without loss of data. The resulting object might show inconsistencies.
A 3-dimensional polyhedron in RDF^3 defined as the convex hull of 120 vertices

>>> W = CoxeterGroup(['I',Integer(7)])
>>> W.permutahedron()
A 2-dimensional polyhedron in AA^2 defined as the convex hull of 14 vertices
>>> W.permutahedron(base_ring=RDF)
A 2-dimensional polyhedron in RDF^2 defined as the convex hull of 14 vertices

>>> W = ReflectionGroup(['A',Integer(3)])                          # optional - gap3
>>> W.permutahedron()                                     # optional - gap3
A 3-dimensional polyhedron in QQ^3 defined as the convex hull
of 24 vertices

>>> W = ReflectionGroup(['A',Integer(3)],['B',Integer(2)])                  # optional - gap3
>>> W.permutahedron()                                     # optional - gap3
A 5-dimensional polyhedron in QQ^5 defined as the convex hull of 192 vertices
../../_images/finite_coxeter_groups-1.svg
reflections_from_w0()[source]

Return the reflections of self using the inversion set of w_0.

EXAMPLES:

sage: WeylGroup(['A',2]).reflections_from_w0()
[
[0 1 0]  [0 0 1]  [1 0 0]
[1 0 0]  [0 1 0]  [0 0 1]
[0 0 1], [1 0 0], [0 1 0]
]

sage: WeylGroup(['A',3]).reflections_from_w0()
[
[0 1 0 0]  [0 0 1 0]  [1 0 0 0]  [0 0 0 1]  [1 0 0 0]  [1 0 0 0]
[1 0 0 0]  [0 1 0 0]  [0 0 1 0]  [0 1 0 0]  [0 0 0 1]  [0 1 0 0]
[0 0 1 0]  [1 0 0 0]  [0 1 0 0]  [0 0 1 0]  [0 0 1 0]  [0 0 0 1]
[0 0 0 1], [0 0 0 1], [0 0 0 1], [1 0 0 0], [0 1 0 0], [0 0 1 0]
]

>>> from sage.all import *
>>> WeylGroup(['A',Integer(2)]).reflections_from_w0()
[
[0 1 0]  [0 0 1]  [1 0 0]
[1 0 0]  [0 1 0]  [0 0 1]
[0 0 1], [1 0 0], [0 1 0]
]

>>> WeylGroup(['A',Integer(3)]).reflections_from_w0()
[
[0 1 0 0]  [0 0 1 0]  [1 0 0 0]  [0 0 0 1]  [1 0 0 0]  [1 0 0 0]
[1 0 0 0]  [0 1 0 0]  [0 0 1 0]  [0 1 0 0]  [0 0 0 1]  [0 1 0 0]
[0 0 1 0]  [1 0 0 0]  [0 1 0 0]  [0 0 1 0]  [0 0 1 0]  [0 0 0 1]
[0 0 0 1], [0 0 0 1], [0 0 0 1], [1 0 0 0], [0 1 0 0], [0 0 1 0]
]
shard_poset(side='right')[source]

Return the shard intersection order attached to \(W\).

This is a lattice structure on \(W\), introduced in [Rea2009]. It contains the noncrossing partition lattice, as the induced lattice on the subset of \(c\)-sortable elements.

The partial order is given by simultaneous inclusion of inversion sets and subgroups attached to every element.

The precise description used here can be found in [STW2018].

Another implementation for the symmetric groups is available as shard_poset().

See also

bhz_poset(), bruhat_poset(), weak_poset()

EXAMPLES:

sage: W = CoxeterGroup(['A',3], base_ring=ZZ)
sage: SH = W.shard_poset(); SH
Finite lattice containing 24 elements
sage: SH.is_graded()
True
sage: SH.characteristic_polynomial()
q^3 - 11*q^2 + 23*q - 13
sage: SH.f_polynomial()
34*q^3 + 22*q^2 + q

>>> from sage.all import *
>>> W = CoxeterGroup(['A',Integer(3)], base_ring=ZZ)
>>> SH = W.shard_poset(); SH
Finite lattice containing 24 elements
>>> SH.is_graded()
True
>>> SH.characteristic_polynomial()
q^3 - 11*q^2 + 23*q - 13
>>> SH.f_polynomial()
34*q^3 + 22*q^2 + q
w0()[source]

Return the longest element of self.

This attribute is deprecated, use long_element() instead.

EXAMPLES:

sage: D8 = FiniteCoxeterGroups().example(8)
sage: D8.w0
(1, 2, 1, 2, 1, 2, 1, 2)
sage: D3 = FiniteCoxeterGroups().example(3)
sage: D3.w0
(1, 2, 1)

>>> from sage.all import *
>>> D8 = FiniteCoxeterGroups().example(Integer(8))
>>> D8.w0
(1, 2, 1, 2, 1, 2, 1, 2)
>>> D3 = FiniteCoxeterGroups().example(Integer(3))
>>> D3.w0
(1, 2, 1)
weak_lattice(side='right', facade=False)[source]

INPUT:

  • side – “left”, “right”, or “twosided” (default: 'right')

  • facade – boolean (default: False)

Returns the left (resp. right) poset for weak order. In this poset, \(u\) is smaller than \(v\) if some reduced word of \(u\) is a right (resp. left) factor of some reduced word of \(v\).

See also

bhz_poset(), bruhat_poset(), shard_poset()

EXAMPLES:

sage: W = WeylGroup(["A", 2])
sage: P = W.weak_poset()
sage: P
Finite lattice containing 6 elements
sage: P.show()

>>> from sage.all import *
>>> W = WeylGroup(["A", Integer(2)])
>>> P = W.weak_poset()
>>> P
Finite lattice containing 6 elements
>>> P.show()

This poset is in fact a lattice:

sage: W = WeylGroup(["B", 3])
sage: P = W.weak_poset(side = "left")
sage: P.is_lattice()
True

>>> from sage.all import *
>>> W = WeylGroup(["B", Integer(3)])
>>> P = W.weak_poset(side = "left")
>>> P.is_lattice()
True

so this method has an alias weak_lattice():

sage: W.weak_lattice(side = "left") is W.weak_poset(side = "left")
True

>>> from sage.all import *
>>> W.weak_lattice(side = "left") is W.weak_poset(side = "left")
True

As a bonus feature, one can create the left-right weak poset:

sage: W = WeylGroup(["A",2])
sage: P = W.weak_poset(side = "twosided")
sage: P.show()
sage: len(P.hasse_diagram().edges(sort=False))
8

>>> from sage.all import *
>>> W = WeylGroup(["A",Integer(2)])
>>> P = W.weak_poset(side = "twosided")
>>> P.show()
>>> len(P.hasse_diagram().edges(sort=False))
8

This is the transitive closure of the union of left and right order. In this poset, \(u\) is smaller than \(v\) if some reduced word of \(u\) is a factor of some reduced word of \(v\). Note that this is not a lattice:

sage: P.is_lattice()
False

>>> from sage.all import *
>>> P.is_lattice()
False

By default, the elements of \(P\) are aware of that they belong to \(P\):

sage: P.an_element().parent()
Finite poset containing 6 elements

>>> from sage.all import *
>>> P.an_element().parent()
Finite poset containing 6 elements

If instead one wants the elements to be plain elements of the Coxeter group, one can use the facade option:

sage: P = W.weak_poset(facade = True)
sage: P.an_element().parent()
Weyl Group of type ['A', 2] (as a matrix group acting on the ambient space)

>>> from sage.all import *
>>> P = W.weak_poset(facade = True)
>>> P.an_element().parent()
Weyl Group of type ['A', 2] (as a matrix group acting on the ambient space)

See also

Poset() for more on posets and facade posets.

Todo

  • Use the symmetric group in the examples (for nicer output), and print the edges for a stronger test.

  • The constructed poset should be lazy, in order to handle large / infinite Coxeter groups.

weak_poset(side='right', facade=False)[source]

INPUT:

  • side – “left”, “right”, or “twosided” (default: 'right')

  • facade – boolean (default: False)

Returns the left (resp. right) poset for weak order. In this poset, \(u\) is smaller than \(v\) if some reduced word of \(u\) is a right (resp. left) factor of some reduced word of \(v\).

See also

bhz_poset(), bruhat_poset(), shard_poset()

EXAMPLES:

sage: W = WeylGroup(["A", 2])
sage: P = W.weak_poset()
sage: P
Finite lattice containing 6 elements
sage: P.show()

>>> from sage.all import *
>>> W = WeylGroup(["A", Integer(2)])
>>> P = W.weak_poset()
>>> P
Finite lattice containing 6 elements
>>> P.show()

This poset is in fact a lattice:

sage: W = WeylGroup(["B", 3])
sage: P = W.weak_poset(side = "left")
sage: P.is_lattice()
True

>>> from sage.all import *
>>> W = WeylGroup(["B", Integer(3)])
>>> P = W.weak_poset(side = "left")
>>> P.is_lattice()
True

so this method has an alias weak_lattice():

sage: W.weak_lattice(side = "left") is W.weak_poset(side = "left")
True

>>> from sage.all import *
>>> W.weak_lattice(side = "left") is W.weak_poset(side = "left")
True

As a bonus feature, one can create the left-right weak poset:

sage: W = WeylGroup(["A",2])
sage: P = W.weak_poset(side = "twosided")
sage: P.show()
sage: len(P.hasse_diagram().edges(sort=False))
8

>>> from sage.all import *
>>> W = WeylGroup(["A",Integer(2)])
>>> P = W.weak_poset(side = "twosided")
>>> P.show()
>>> len(P.hasse_diagram().edges(sort=False))
8

This is the transitive closure of the union of left and right order. In this poset, \(u\) is smaller than \(v\) if some reduced word of \(u\) is a factor of some reduced word of \(v\). Note that this is not a lattice:

sage: P.is_lattice()
False

>>> from sage.all import *
>>> P.is_lattice()
False

By default, the elements of \(P\) are aware of that they belong to \(P\):

sage: P.an_element().parent()
Finite poset containing 6 elements

>>> from sage.all import *
>>> P.an_element().parent()
Finite poset containing 6 elements

If instead one wants the elements to be plain elements of the Coxeter group, one can use the facade option:

sage: P = W.weak_poset(facade = True)
sage: P.an_element().parent()
Weyl Group of type ['A', 2] (as a matrix group acting on the ambient space)

>>> from sage.all import *
>>> P = W.weak_poset(facade = True)
>>> P.an_element().parent()
Weyl Group of type ['A', 2] (as a matrix group acting on the ambient space)

See also

Poset() for more on posets and facade posets.

Todo

  • Use the symmetric group in the examples (for nicer output), and print the edges for a stronger test.

  • The constructed poset should be lazy, in order to handle large / infinite Coxeter groups.

extra_super_categories()[source]

EXAMPLES:

sage: CoxeterGroups().Finite().super_categories()
[Category of finite generalized Coxeter groups,
 Category of Coxeter groups]

>>> from sage.all import *
>>> CoxeterGroups().Finite().super_categories()
[Category of finite generalized Coxeter groups,
 Category of Coxeter groups]