Finite Coxeter Groups

class sage.categories.finite_coxeter_groups.FiniteCoxeterGroups(base_category)

Bases: sage.categories.category_with_axiom.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

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)

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
class ElementMethods
bruhat_upper_covers()

Returns 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]]

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()

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
coxeter_knuth_graph()

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()
[(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()
[((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()
[(1, 3), (3, 1)]
sage: D.edges()
[]
coxeter_knuth_neighbor(w)

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)}
is_coxeter_element()

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
class ParentMethods

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

bhz_poset()

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 succintly 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 \(W_I\)).

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
bruhat_poset(facade=False)

Return the Bruhat poset of self.

EXAMPLES:

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

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

sage: 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)

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)

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 (optional, 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
codegrees()

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)
degrees()

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)
inversion_sequence(word)

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]]
is_real()

Return True since self is a real reflection group.

EXAMPLES:

sage: CoxeterGroup(['F',4]).is_real()
True
sage: CoxeterGroup(['H',4]).is_real()
True
long_element(index_set=None, as_word=False)

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)

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]
m_cambrian_lattice(c, m=1, on_roots=False)

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)
  • \(m\) – a positive integer (optional, default 1)
  • on_roots (optional, 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
permutahedron(point=None, base_ring=None)

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 is \((1, 1, 1, \ldots)\))
  • 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
../../_images/finite_coxeter_groups-1.svg
reflections_from_w0()

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]
]
shard_poset(side='right')

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().

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
w0()

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)
weak_lattice(side='right', facade=False)

INPUT:

  • side – “left”, “right”, or “twosided” (default: “right”)
  • facade – a 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 \(v\).

EXAMPLES:

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

so this method has an alias weak_lattice():

sage: 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())
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

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

sage: 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)

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)

INPUT:

  • side – “left”, “right”, or “twosided” (default: “right”)
  • facade – a 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 \(v\).

EXAMPLES:

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

so this method has an alias weak_lattice():

sage: 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())
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

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

sage: 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)

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()

EXAMPLES:

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