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 - selfin 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 - wunder 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 - selfhas 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_elementsis 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 - 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 - 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 - facadeoption:- 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_rootsis- True, the lattice is realized on roots rather than on reflections. In order for this to work, the ElementMethod- reflection_to_rootmust 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 - wordin indices of simple generators of- self.- If - wordcorresponds 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 - Truesince- selfis 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_rootsis- True, the lattice is realized on roots rather than on reflections. In order for this to work, the ElementMethod- reflection_to_rootmust 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 - pointin the weight basis under the action of- selfon 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 - RDFwill 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 
 - reflections_from_w0()[source]¶
- Return the reflections of - selfusing 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 - 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 - 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 - facadeoption:- 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 - 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 - facadeoption:- 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]