Finite complex reflection groups¶
Let \(V\) be a finite-dimensional complex vector space. A reflection of \(V\) is an operator \(r \in \operatorname{GL}(V)\) that has finite order and fixes pointwise a hyperplane in \(V\).
For more definitions and classification types of finite complex reflection groups, see Wikipedia article Complex_reflection_group.
The point of entry to work with reflection groups is ReflectionGroup()
which can be used with finite Cartan-Killing types:
sage: ReflectionGroup(['A',2])
Irreducible real reflection group of rank 2 and type A2
sage: ReflectionGroup(['F',4])
Irreducible real reflection group of rank 4 and type F4
sage: ReflectionGroup(['H',3])
Irreducible real reflection group of rank 3 and type H3
>>> from sage.all import *
>>> ReflectionGroup(['A',Integer(2)])
Irreducible real reflection group of rank 2 and type A2
>>> ReflectionGroup(['F',Integer(4)])
Irreducible real reflection group of rank 4 and type F4
>>> ReflectionGroup(['H',Integer(3)])
Irreducible real reflection group of rank 3 and type H3
or with Shephard-Todd types:
sage: ReflectionGroup((1,1,3))
Irreducible real reflection group of rank 2 and type A2
sage: ReflectionGroup((2,1,3))
Irreducible real reflection group of rank 3 and type B3
sage: ReflectionGroup((3,1,3))
Irreducible complex reflection group of rank 3 and type G(3,1,3)
sage: ReflectionGroup((4,2,3))
Irreducible complex reflection group of rank 3 and type G(4,2,3)
sage: ReflectionGroup(4)
Irreducible complex reflection group of rank 2 and type ST4
sage: ReflectionGroup(31)
Irreducible complex reflection group of rank 4 and type ST31
>>> from sage.all import *
>>> ReflectionGroup((Integer(1),Integer(1),Integer(3)))
Irreducible real reflection group of rank 2 and type A2
>>> ReflectionGroup((Integer(2),Integer(1),Integer(3)))
Irreducible real reflection group of rank 3 and type B3
>>> ReflectionGroup((Integer(3),Integer(1),Integer(3)))
Irreducible complex reflection group of rank 3 and type G(3,1,3)
>>> ReflectionGroup((Integer(4),Integer(2),Integer(3)))
Irreducible complex reflection group of rank 3 and type G(4,2,3)
>>> ReflectionGroup(Integer(4))
Irreducible complex reflection group of rank 2 and type ST4
>>> ReflectionGroup(Integer(31))
Irreducible complex reflection group of rank 4 and type ST31
Also reducible types are allowed using concatenation:
sage: ReflectionGroup(['A',3],(4,2,3))
Reducible complex reflection group of rank 6 and type A3 x G(4,2,3)
>>> from sage.all import *
>>> ReflectionGroup(['A',Integer(3)],(Integer(4),Integer(2),Integer(3)))
Reducible complex reflection group of rank 6 and type A3 x G(4,2,3)
Some special cases also occur, among them are:
sage: W = ReflectionGroup((2,2,2)); W
Reducible real reflection group of rank 2 and type A1 x A1
sage: W = ReflectionGroup((2,2,3)); W
Irreducible real reflection group of rank 3 and type A3
>>> from sage.all import *
>>> W = ReflectionGroup((Integer(2),Integer(2),Integer(2))); W
Reducible real reflection group of rank 2 and type A1 x A1
>>> W = ReflectionGroup((Integer(2),Integer(2),Integer(3))); W
Irreducible real reflection group of rank 3 and type A3
Warning
Uses the GAP3 package Chevie which is available as an
experimental package (installed by sage -i gap3) or to
download by hand from Jean Michel’s website.
A guided tour¶
We start with the example type \(B_2\):
sage: W = ReflectionGroup(['B',2]); W
Irreducible real reflection group of rank 2 and type B2
>>> from sage.all import *
>>> W = ReflectionGroup(['B',Integer(2)]); W
Irreducible real reflection group of rank 2 and type B2
Most importantly, observe that the group elements are usually represented by permutations of the roots:
sage: for w in W: print(w)
()
(1,3)(2,6)(5,7)
(1,5)(2,4)(6,8)
(1,7,5,3)(2,4,6,8)
(1,3,5,7)(2,8,6,4)
(2,8)(3,7)(4,6)
(1,7)(3,5)(4,8)
(1,5)(2,6)(3,7)(4,8)
>>> from sage.all import *
>>> for w in W: print(w)
()
(1,3)(2,6)(5,7)
(1,5)(2,4)(6,8)
(1,7,5,3)(2,4,6,8)
(1,3,5,7)(2,8,6,4)
(2,8)(3,7)(4,6)
(1,7)(3,5)(4,8)
(1,5)(2,6)(3,7)(4,8)
This has the drawback that one can hardly see anything. Usually, one would look at elements with either of the following methods:
sage: for w in W: w.reduced_word()
[]
[2]
[1]
[1, 2]
[2, 1]
[2, 1, 2]
[1, 2, 1]
[1, 2, 1, 2]
sage: for w in W: w.reduced_word_in_reflections()
[]
[2]
[1]
[1, 2]
[1, 4]
[3]
[4]
[1, 3]
sage: for w in W: w.reduced_word(); w.to_matrix(); print("")
[]
[1 0]
[0 1]
[2]
[ 1 1]
[ 0 -1]
[1]
[-1 0]
[ 2 1]
[1, 2]
[-1 -1]
[ 2 1]
[2, 1]
[ 1 1]
[-2 -1]
[2, 1, 2]
[ 1 0]
[-2 -1]
[1, 2, 1]
[-1 -1]
[ 0 1]
[1, 2, 1, 2]
[-1 0]
[ 0 -1]
>>> from sage.all import *
>>> for w in W: w.reduced_word()
[]
[2]
[1]
[1, 2]
[2, 1]
[2, 1, 2]
[1, 2, 1]
[1, 2, 1, 2]
>>> for w in W: w.reduced_word_in_reflections()
[]
[2]
[1]
[1, 2]
[1, 4]
[3]
[4]
[1, 3]
>>> for w in W: w.reduced_word(); w.to_matrix(); print("")
[]
[1 0]
[0 1]
<BLANKLINE>
[2]
[ 1 1]
[ 0 -1]
<BLANKLINE>
[1]
[-1 0]
[ 2 1]
<BLANKLINE>
[1, 2]
[-1 -1]
[ 2 1]
<BLANKLINE>
[2, 1]
[ 1 1]
[-2 -1]
<BLANKLINE>
[2, 1, 2]
[ 1 0]
[-2 -1]
<BLANKLINE>
[1, 2, 1]
[-1 -1]
[ 0 1]
<BLANKLINE>
[1, 2, 1, 2]
[-1 0]
[ 0 -1]
<BLANKLINE>
The standard references for actions of complex reflection groups have the matrices acting on the right, so:
sage: W.simple_reflection(1).to_matrix()
[-1 0]
[ 2 1]
>>> from sage.all import *
>>> W.simple_reflection(Integer(1)).to_matrix()
[-1 0]
[ 2 1]
sends the simple root \(\alpha_0\), or (1,0) in vector notation, to
its negative, while sending \(\alpha_1\) to \(2\alpha_0+\alpha_1\).
Todo
properly provide root systems for real reflection groups
element class should be unique to be able to work with large groups without creating elements multiple times
is_shephard_group,is_generalized_coxeter_groupexponents and coexponents
coinvariant ring:
fake degrees from Torsten Hoge
operation of linear characters on all characters
harmonic polynomials
linear forms for hyperplanes
field of definition
intersection lattice and characteristic polynomial:
X = [ alpha(t) for t in W.distinguished_reflections() ] X = Matrix(CF,X).transpose() Y = Matroid(X)
linear characters
permutation pi on irreducibles
hyperplane orbits (76.13 in Gap Manual)
improve invariant_form with a code similar to the one in
reflection_group_real.pyadd a method
reflection_to_rootordistinguished_reflection_to_positive_rootdiagrams in ASCII-art (76.15)
standard (BMR) presentations
character table directly from Chevie
GenericOrder(76.20),TorusOrder(76.21)correct fundamental invariants for \(G_34\), check the others
copy hardcoded data (degrees, invariants, braid relations…) into sage
add other hardcoded data from the tables in chevie (location is SAGEDIR/local/gap3/gap-jm5-2015-02-01/gap3/pkg/chevie/tbl): basic derivations, discriminant, …
transfer code for
reduced_word_in_reflectionsinto Gap4 or Sagelist of reduced words for an element
list of reduced words in reflections for an element
Hurwitz action?
is_crystallographic()should be hardcoded
AUTHORS:
Christian Stump (2015): initial version
- class sage.combinat.root_system.reflection_group_complex.ComplexReflectionGroup(W_types, index_set=None, hyperplane_index_set=None, reflection_index_set=None)[source]¶
Bases:
UniqueRepresentation,PermutationGroup_genericA complex reflection group given as a permutation group.
See also
- class Element[source]¶
Bases:
ComplexReflectionGroupElement- conjugacy_class()[source]¶
Return the conjugacy class of
self.EXAMPLES:
sage: W = ReflectionGroup((1,1,3)) sage: for w in W: sorted(w.conjugacy_class()) [()] [(1,3)(2,5)(4,6), (1,4)(2,3)(5,6), (1,5)(2,4)(3,6)] [(1,3)(2,5)(4,6), (1,4)(2,3)(5,6), (1,5)(2,4)(3,6)] [(1,2,6)(3,4,5), (1,6,2)(3,5,4)] [(1,2,6)(3,4,5), (1,6,2)(3,5,4)] [(1,3)(2,5)(4,6), (1,4)(2,3)(5,6), (1,5)(2,4)(3,6)]
>>> from sage.all import * >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(3))) >>> for w in W: sorted(w.conjugacy_class()) [()] [(1,3)(2,5)(4,6), (1,4)(2,3)(5,6), (1,5)(2,4)(3,6)] [(1,3)(2,5)(4,6), (1,4)(2,3)(5,6), (1,5)(2,4)(3,6)] [(1,2,6)(3,4,5), (1,6,2)(3,5,4)] [(1,2,6)(3,4,5), (1,6,2)(3,5,4)] [(1,3)(2,5)(4,6), (1,4)(2,3)(5,6), (1,5)(2,4)(3,6)]
- conjugacy_class_representative()[source]¶
Return a representative of the conjugacy class of
self.EXAMPLES:
sage: W = ReflectionGroup((1,1,3)) sage: for w in W: ....: print('%s %s'%(w.reduced_word(), w.conjugacy_class_representative().reduced_word())) [] [] [2] [1] [1] [1] [1, 2] [1, 2] [2, 1] [1, 2] [1, 2, 1] [1]
>>> from sage.all import * >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(3))) >>> for w in W: ... print('%s %s'%(w.reduced_word(), w.conjugacy_class_representative().reduced_word())) [] [] [2] [1] [1] [1] [1, 2] [1, 2] [2, 1] [1, 2] [1, 2, 1] [1]
- reflection_length(in_unitary_group=False)[source]¶
Return the reflection length of
self.This is the minimal numbers of reflections needed to obtain
self.INPUT:
in_unitary_group– boolean (default:False); ifTrue, the reflection length is computed in the unitary group which is the dimension of the move space ofself
EXAMPLES:
sage: W = ReflectionGroup((1,1,3)) sage: sorted([t.reflection_length() for t in W]) [0, 1, 1, 1, 2, 2] sage: W = ReflectionGroup((2,1,2)) sage: sorted([t.reflection_length() for t in W]) [0, 1, 1, 1, 1, 2, 2, 2] sage: W = ReflectionGroup((2,2,2)) sage: sorted([t.reflection_length() for t in W]) [0, 1, 1, 2] sage: W = ReflectionGroup((3,1,2)) sage: sorted([t.reflection_length() for t in W]) [0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]
>>> from sage.all import * >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(3))) >>> sorted([t.reflection_length() for t in W]) [0, 1, 1, 1, 2, 2] >>> W = ReflectionGroup((Integer(2),Integer(1),Integer(2))) >>> sorted([t.reflection_length() for t in W]) [0, 1, 1, 1, 1, 2, 2, 2] >>> W = ReflectionGroup((Integer(2),Integer(2),Integer(2))) >>> sorted([t.reflection_length() for t in W]) [0, 1, 1, 2] >>> W = ReflectionGroup((Integer(3),Integer(1),Integer(2))) >>> sorted([t.reflection_length() for t in W]) [0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]
- apply_vector_field(f, vf=None)[source]¶
Return a rational function obtained by applying the vector field
vfto the rational functionf.If
vfis not given, the primitive vector field is used.EXAMPLES:
sage: W = ReflectionGroup(['A',2]) sage: for x in W.primitive_vector_field()[0].parent().gens(): ....: print(W.apply_vector_field(x)) 3*x1/(6*x0^2 - 6*x0*x1 - 12*x1^2) 1/(6*x0^2 - 6*x0*x1 - 12*x1^2)
>>> from sage.all import * >>> W = ReflectionGroup(['A',Integer(2)]) >>> for x in W.primitive_vector_field()[Integer(0)].parent().gens(): ... print(W.apply_vector_field(x)) 3*x1/(6*x0^2 - 6*x0*x1 - 12*x1^2) 1/(6*x0^2 - 6*x0*x1 - 12*x1^2)
- braid_relations()[source]¶
Return the braid relations of
self.EXAMPLES:
sage: W = ReflectionGroup((1,1,3)) sage: W.braid_relations() [[[1, 2, 1], [2, 1, 2]]] sage: W = ReflectionGroup((2,1,3)) sage: W.braid_relations() [[[1, 2, 1, 2], [2, 1, 2, 1]], [[1, 3], [3, 1]], [[2, 3, 2], [3, 2, 3]]] sage: W = ReflectionGroup((2,2,3)) sage: W.braid_relations() [[[1, 2, 1], [2, 1, 2]], [[1, 3], [3, 1]], [[2, 3, 2], [3, 2, 3]]]
>>> from sage.all import * >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(3))) >>> W.braid_relations() [[[1, 2, 1], [2, 1, 2]]] >>> W = ReflectionGroup((Integer(2),Integer(1),Integer(3))) >>> W.braid_relations() [[[1, 2, 1, 2], [2, 1, 2, 1]], [[1, 3], [3, 1]], [[2, 3, 2], [3, 2, 3]]] >>> W = ReflectionGroup((Integer(2),Integer(2),Integer(3))) >>> W.braid_relations() [[[1, 2, 1], [2, 1, 2]], [[1, 3], [3, 1]], [[2, 3, 2], [3, 2, 3]]]
- cartan_matrix()[source]¶
Return the Cartan matrix associated with
self.If
selfis crystallographic, the returned Cartan matrix is an instance ofCartanMatrix, and a normal matrix otherwise.Let \(s_1, \ldots, s_n\) be a set of reflections which generate
selfwith associated simple roots \(s_1,\ldots,s_n\) and simple coroots \(s^\vee_i\). Then the Cartan matrix \(C = (c_{ij})\) is given by \(s^\vee_i(s_j)\). The Cartan matrix completely determines the reflection representation if the \(s_i\) are linearly independent.EXAMPLES:
sage: ReflectionGroup(['A',4]).cartan_matrix() [ 2 -1 0 0] [-1 2 -1 0] [ 0 -1 2 -1] [ 0 0 -1 2] sage: ReflectionGroup(['H',4]).cartan_matrix() [ 2 E(5)^2 + E(5)^3 0 0] [E(5)^2 + E(5)^3 2 -1 0] [ 0 -1 2 -1] [ 0 0 -1 2] sage: ReflectionGroup(4).cartan_matrix() [-2*E(3) - E(3)^2 E(3)^2] [ -E(3)^2 -2*E(3) - E(3)^2] sage: ReflectionGroup((4,2,2)).cartan_matrix() [ 2 -2*E(4) -2] [ E(4) 2 1 - E(4)] [ -1 1 + E(4) 2]
>>> from sage.all import * >>> ReflectionGroup(['A',Integer(4)]).cartan_matrix() [ 2 -1 0 0] [-1 2 -1 0] [ 0 -1 2 -1] [ 0 0 -1 2] >>> ReflectionGroup(['H',Integer(4)]).cartan_matrix() [ 2 E(5)^2 + E(5)^3 0 0] [E(5)^2 + E(5)^3 2 -1 0] [ 0 -1 2 -1] [ 0 0 -1 2] >>> ReflectionGroup(Integer(4)).cartan_matrix() [-2*E(3) - E(3)^2 E(3)^2] [ -E(3)^2 -2*E(3) - E(3)^2] >>> ReflectionGroup((Integer(4),Integer(2),Integer(2))).cartan_matrix() [ 2 -2*E(4) -2] [ E(4) 2 1 - E(4)] [ -1 1 + E(4) 2]
- codegrees()[source]¶
Return the codegrees of
selfordered within each irreducible component ofself.EXAMPLES:
sage: W = ReflectionGroup((1,1,4)) sage: W.codegrees() (2, 1, 0) sage: W = ReflectionGroup((2,1,4)) sage: W.codegrees() (6, 4, 2, 0) sage: W = ReflectionGroup((4,1,4)) sage: W.codegrees() (12, 8, 4, 0) sage: W = ReflectionGroup((4,2,4)) sage: W.codegrees() (12, 8, 4, 0) sage: W = ReflectionGroup((4,4,4)) sage: W.codegrees() (8, 8, 4, 0) sage: W = ReflectionGroup((1,1,4), (3,1,2)) sage: W.codegrees() (2, 1, 0, 3, 0) sage: W = ReflectionGroup((1,1,4), (6,1,12), 23) sage: W.codegrees() (2, 1, 0, 66, 60, 54, 48, 42, 36, 30, 24, 18, 12, 6, 0, 8, 4, 0)
>>> from sage.all import * >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(4))) >>> W.codegrees() (2, 1, 0) >>> W = ReflectionGroup((Integer(2),Integer(1),Integer(4))) >>> W.codegrees() (6, 4, 2, 0) >>> W = ReflectionGroup((Integer(4),Integer(1),Integer(4))) >>> W.codegrees() (12, 8, 4, 0) >>> W = ReflectionGroup((Integer(4),Integer(2),Integer(4))) >>> W.codegrees() (12, 8, 4, 0) >>> W = ReflectionGroup((Integer(4),Integer(4),Integer(4))) >>> W.codegrees() (8, 8, 4, 0) >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(4)), (Integer(3),Integer(1),Integer(2))) >>> W.codegrees() (2, 1, 0, 3, 0) >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(4)), (Integer(6),Integer(1),Integer(12)), Integer(23)) >>> W.codegrees() (2, 1, 0, 66, 60, 54, 48, 42, 36, 30, 24, 18, 12, 6, 0, 8, 4, 0)
- conjugacy_classes()[source]¶
Return the conjugacy classes of
self.EXAMPLES:
sage: W = ReflectionGroup((1,1,3)) sage: for C in W.conjugacy_classes(): sorted(C) [()] [(1,3)(2,5)(4,6), (1,4)(2,3)(5,6), (1,5)(2,4)(3,6)] [(1,2,6)(3,4,5), (1,6,2)(3,5,4)] sage: W = ReflectionGroup((1,1,4)) sage: sum(len(C) for C in W.conjugacy_classes()) == W.cardinality() True sage: W = ReflectionGroup((3,1,2)) sage: sum(len(C) for C in W.conjugacy_classes()) == W.cardinality() True sage: W = ReflectionGroup(23) sage: sum(len(C) for C in W.conjugacy_classes()) == W.cardinality() True
>>> from sage.all import * >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(3))) >>> for C in W.conjugacy_classes(): sorted(C) [()] [(1,3)(2,5)(4,6), (1,4)(2,3)(5,6), (1,5)(2,4)(3,6)] [(1,2,6)(3,4,5), (1,6,2)(3,5,4)] >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(4))) >>> sum(len(C) for C in W.conjugacy_classes()) == W.cardinality() True >>> W = ReflectionGroup((Integer(3),Integer(1),Integer(2))) >>> sum(len(C) for C in W.conjugacy_classes()) == W.cardinality() True >>> W = ReflectionGroup(Integer(23)) >>> sum(len(C) for C in W.conjugacy_classes()) == W.cardinality() True
- conjugacy_classes_representatives()[source]¶
Return the shortest representatives of the conjugacy classes of
self.EXAMPLES:
sage: W = ReflectionGroup((1,1,3)) sage: [w.reduced_word() for w in W.conjugacy_classes_representatives()] [[], [1], [1, 2]] sage: W = ReflectionGroup((1,1,4)) sage: [w.reduced_word() for w in W.conjugacy_classes_representatives()] [[], [1], [1, 3], [1, 2], [1, 3, 2]] sage: W = ReflectionGroup((3,1,2)) sage: [w.reduced_word() for w in W.conjugacy_classes_representatives()] [[], [1], [1, 1], [2, 1, 2, 1], [2, 1, 2, 1, 1], [2, 1, 1, 2, 1, 1], [2], [1, 2], [1, 1, 2]] sage: W = ReflectionGroup(23) sage: [w.reduced_word() for w in W.conjugacy_classes_representatives()] [[], [1], [1, 2], [1, 3], [2, 3], [1, 2, 3], [1, 2, 1, 2], [1, 2, 1, 2, 3], [1, 2, 1, 2, 3, 2, 1, 2, 3], [1, 2, 1, 2, 1, 3, 2, 1, 2, 1, 3, 2, 1, 2, 3]]
>>> from sage.all import * >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(3))) >>> [w.reduced_word() for w in W.conjugacy_classes_representatives()] [[], [1], [1, 2]] >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(4))) >>> [w.reduced_word() for w in W.conjugacy_classes_representatives()] [[], [1], [1, 3], [1, 2], [1, 3, 2]] >>> W = ReflectionGroup((Integer(3),Integer(1),Integer(2))) >>> [w.reduced_word() for w in W.conjugacy_classes_representatives()] [[], [1], [1, 1], [2, 1, 2, 1], [2, 1, 2, 1, 1], [2, 1, 1, 2, 1, 1], [2], [1, 2], [1, 1, 2]] >>> W = ReflectionGroup(Integer(23)) >>> [w.reduced_word() for w in W.conjugacy_classes_representatives()] [[], [1], [1, 2], [1, 3], [2, 3], [1, 2, 3], [1, 2, 1, 2], [1, 2, 1, 2, 3], [1, 2, 1, 2, 3, 2, 1, 2, 3], [1, 2, 1, 2, 1, 3, 2, 1, 2, 1, 3, 2, 1, 2, 3]]
- coxeter_number(chi=None)[source]¶
Return the Coxeter number associated to the irreducible character chi of the reflection group
self.The Coxeter number of a complex reflection group \(W\) is the trace in a character \(\chi\) of \(\sum_t (Id - t)\), where \(t\) runs over all reflections. The result is always an integer.
When \(\chi\) is the reflection representation, the Coxeter number is equal to \(\frac{N + N^*}{n}\) where \(N\) is the number of reflections, \(N^*\) is the number of reflection hyperplanes, and \(n\) is the rank of \(W\). If \(W\) is further well-generated, the Coxeter number is equal to the highest degree d_n and to the order of a Coxeter element \(c\) of \(W\).
EXAMPLES:
sage: W = ReflectionGroup(["H",4]) sage: W.coxeter_number() 30 sage: all(W.coxeter_number(chi).is_integer() ....: for chi in W.irreducible_characters()) True sage: W = ReflectionGroup(14) sage: W.coxeter_number() 24
>>> from sage.all import * >>> W = ReflectionGroup(["H",Integer(4)]) >>> W.coxeter_number() 30 >>> all(W.coxeter_number(chi).is_integer() ... for chi in W.irreducible_characters()) True >>> W = ReflectionGroup(Integer(14)) >>> W.coxeter_number() 24
- degrees()[source]¶
Return the degrees of
selfordered within each irreducible component ofself.EXAMPLES:
sage: W = ReflectionGroup((1,1,4)) sage: W.degrees() (2, 3, 4) sage: W = ReflectionGroup((2,1,4)) sage: W.degrees() (2, 4, 6, 8) sage: W = ReflectionGroup((4,1,4)) sage: W.degrees() (4, 8, 12, 16) sage: W = ReflectionGroup((4,2,4)) sage: W.degrees() (4, 8, 8, 12) sage: W = ReflectionGroup((4,4,4)) sage: W.degrees() (4, 4, 8, 12)
>>> from sage.all import * >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(4))) >>> W.degrees() (2, 3, 4) >>> W = ReflectionGroup((Integer(2),Integer(1),Integer(4))) >>> W.degrees() (2, 4, 6, 8) >>> W = ReflectionGroup((Integer(4),Integer(1),Integer(4))) >>> W.degrees() (4, 8, 12, 16) >>> W = ReflectionGroup((Integer(4),Integer(2),Integer(4))) >>> W.degrees() (4, 8, 8, 12) >>> W = ReflectionGroup((Integer(4),Integer(4),Integer(4))) >>> W.degrees() (4, 4, 8, 12)
Examples of reducible types:
sage: W = ReflectionGroup((1,1,4), (3,1,2)); W Reducible complex reflection group of rank 5 and type A3 x G(3,1,2) sage: W.degrees() (2, 3, 4, 3, 6) sage: W = ReflectionGroup((1,1,4), (6,1,12), 23) sage: W.degrees() (2, 3, 4, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 2, 6, 10)
>>> from sage.all import * >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(4)), (Integer(3),Integer(1),Integer(2))); W Reducible complex reflection group of rank 5 and type A3 x G(3,1,2) >>> W.degrees() (2, 3, 4, 3, 6) >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(4)), (Integer(6),Integer(1),Integer(12)), Integer(23)) >>> W.degrees() (2, 3, 4, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 2, 6, 10)
- discriminant()[source]¶
Return the discriminant of
selfin the polynomial ring on which the group acts.This is the product
\[\prod_H \alpha_H^{e_H},\]where \(\alpha_H\) is the linear form of the hyperplane \(H\) and \(e_H\) is its stabilizer order.
EXAMPLES:
sage: W = ReflectionGroup(['A',2]) sage: W.discriminant() x0^6 - 3*x0^5*x1 - 3/4*x0^4*x1^2 + 13/2*x0^3*x1^3 - 3/4*x0^2*x1^4 - 3*x0*x1^5 + x1^6 sage: W = ReflectionGroup(['B',2]) sage: W.discriminant() x0^6*x1^2 - 6*x0^5*x1^3 + 13*x0^4*x1^4 - 12*x0^3*x1^5 + 4*x0^2*x1^6
>>> from sage.all import * >>> W = ReflectionGroup(['A',Integer(2)]) >>> W.discriminant() x0^6 - 3*x0^5*x1 - 3/4*x0^4*x1^2 + 13/2*x0^3*x1^3 - 3/4*x0^2*x1^4 - 3*x0*x1^5 + x1^6 >>> W = ReflectionGroup(['B',Integer(2)]) >>> W.discriminant() x0^6*x1^2 - 6*x0^5*x1^3 + 13*x0^4*x1^4 - 12*x0^3*x1^5 + 4*x0^2*x1^6
- discriminant_in_invariant_ring(invariants=None)[source]¶
Return the discriminant of
selfin the invariant ring.This is the function \(f\) in the invariants such that \(f(F_1(x), \ldots, F_n(x))\) is the discriminant.
EXAMPLES:
sage: W = ReflectionGroup(['A',3]) sage: W.discriminant_in_invariant_ring() 6*t0^3*t1^2 - 18*t0^4*t2 + 9*t1^4 - 36*t0*t1^2*t2 + 24*t0^2*t2^2 - 8*t2^3 sage: W = ReflectionGroup(['B',3]) sage: W.discriminant_in_invariant_ring() -t0^2*t1^2*t2 + 16*t0^3*t2^2 + 2*t1^3*t2 - 36*t0*t1*t2^2 + 108*t2^3 sage: W = ReflectionGroup(['H',3]) sage: W.discriminant_in_invariant_ring() # long time (-829*E(5) - 1658*E(5)^2 - 1658*E(5)^3 - 829*E(5)^4)*t0^15 + (213700*E(5) + 427400*E(5)^2 + 427400*E(5)^3 + 213700*E(5)^4)*t0^12*t1 + (-22233750*E(5) - 44467500*E(5)^2 - 44467500*E(5)^3 - 22233750*E(5)^4)*t0^9*t1^2 + (438750*E(5) + 877500*E(5)^2 + 877500*E(5)^3 + 438750*E(5)^4)*t0^10*t2 + (1162187500*E(5) + 2324375000*E(5)^2 + 2324375000*E(5)^3 + 1162187500*E(5)^4)*t0^6*t1^3 + (-74250000*E(5) - 148500000*E(5)^2 - 148500000*E(5)^3 - 74250000*E(5)^4)*t0^7*t1*t2 + (-28369140625*E(5) - 56738281250*E(5)^2 - 56738281250*E(5)^3 - 28369140625*E(5)^4)*t0^3*t1^4 + (1371093750*E(5) + 2742187500*E(5)^2 + 2742187500*E(5)^3 + 1371093750*E(5)^4)*t0^4*t1^2*t2 + (1191796875*E(5) + 2383593750*E(5)^2 + 2383593750*E(5)^3 + 1191796875*E(5)^4)*t0^5*t2^2 + (175781250000*E(5) + 351562500000*E(5)^2 + 351562500000*E(5)^3 + 175781250000*E(5)^4)*t1^5 + (131835937500*E(5) + 263671875000*E(5)^2 + 263671875000*E(5)^3 + 131835937500*E(5)^4)*t0*t1^3*t2 + (-100195312500*E(5) - 200390625000*E(5)^2 - 200390625000*E(5)^3 - 100195312500*E(5)^4)*t0^2*t1*t2^2 + (395507812500*E(5) + 791015625000*E(5)^2 + 791015625000*E(5)^3 + 395507812500*E(5)^4)*t2^3
>>> from sage.all import * >>> W = ReflectionGroup(['A',Integer(3)]) >>> W.discriminant_in_invariant_ring() 6*t0^3*t1^2 - 18*t0^4*t2 + 9*t1^4 - 36*t0*t1^2*t2 + 24*t0^2*t2^2 - 8*t2^3 >>> W = ReflectionGroup(['B',Integer(3)]) >>> W.discriminant_in_invariant_ring() -t0^2*t1^2*t2 + 16*t0^3*t2^2 + 2*t1^3*t2 - 36*t0*t1*t2^2 + 108*t2^3 >>> W = ReflectionGroup(['H',Integer(3)]) >>> W.discriminant_in_invariant_ring() # long time (-829*E(5) - 1658*E(5)^2 - 1658*E(5)^3 - 829*E(5)^4)*t0^15 + (213700*E(5) + 427400*E(5)^2 + 427400*E(5)^3 + 213700*E(5)^4)*t0^12*t1 + (-22233750*E(5) - 44467500*E(5)^2 - 44467500*E(5)^3 - 22233750*E(5)^4)*t0^9*t1^2 + (438750*E(5) + 877500*E(5)^2 + 877500*E(5)^3 + 438750*E(5)^4)*t0^10*t2 + (1162187500*E(5) + 2324375000*E(5)^2 + 2324375000*E(5)^3 + 1162187500*E(5)^4)*t0^6*t1^3 + (-74250000*E(5) - 148500000*E(5)^2 - 148500000*E(5)^3 - 74250000*E(5)^4)*t0^7*t1*t2 + (-28369140625*E(5) - 56738281250*E(5)^2 - 56738281250*E(5)^3 - 28369140625*E(5)^4)*t0^3*t1^4 + (1371093750*E(5) + 2742187500*E(5)^2 + 2742187500*E(5)^3 + 1371093750*E(5)^4)*t0^4*t1^2*t2 + (1191796875*E(5) + 2383593750*E(5)^2 + 2383593750*E(5)^3 + 1191796875*E(5)^4)*t0^5*t2^2 + (175781250000*E(5) + 351562500000*E(5)^2 + 351562500000*E(5)^3 + 175781250000*E(5)^4)*t1^5 + (131835937500*E(5) + 263671875000*E(5)^2 + 263671875000*E(5)^3 + 131835937500*E(5)^4)*t0*t1^3*t2 + (-100195312500*E(5) - 200390625000*E(5)^2 - 200390625000*E(5)^3 - 100195312500*E(5)^4)*t0^2*t1*t2^2 + (395507812500*E(5) + 791015625000*E(5)^2 + 791015625000*E(5)^3 + 395507812500*E(5)^4)*t2^3
- distinguished_reflection(i)[source]¶
Return the
i-th distinguished reflection ofself.These are the reflections in
selfacting on the complement of the fixed hyperplane \(H\) as \(\operatorname{exp}(2 \pi i / n)\), where \(n\) is the order of the reflection subgroup fixing \(H\).EXAMPLES:
sage: W = ReflectionGroup((1,1,3)) sage: W.distinguished_reflection(1) (1,4)(2,3)(5,6) sage: W.distinguished_reflection(2) (1,3)(2,5)(4,6) sage: W.distinguished_reflection(3) (1,5)(2,4)(3,6) sage: W = ReflectionGroup((3,1,1),hyperplane_index_set=['a']) sage: W.distinguished_reflection('a') (1,2,3) sage: W = ReflectionGroup((1,1,3),(3,1,2)) sage: for i in range(W.number_of_reflection_hyperplanes()): ....: W.distinguished_reflection(i+1) (1,6)(2,5)(7,8) (1,5)(2,7)(6,8) (3,9,15)(4,10,16)(12,17,23)(14,18,24)(20,25,29)(21,22,26)(27,28,30) (3,11)(4,12)(9,13)(10,14)(15,19)(16,20)(17,21)(18,22)(23,27)(24,28)(25,26)(29,30) (1,7)(2,6)(5,8) (3,19)(4,25)(9,11)(10,17)(12,28)(13,15)(14,30)(16,18)(20,27)(21,29)(22,23)(24,26) (4,21,27)(10,22,28)(11,13,19)(12,14,20)(16,26,30)(17,18,25)(23,24,29) (3,13)(4,24)(9,19)(10,29)(11,15)(12,26)(14,21)(16,23)(17,30)(18,27)(20,22)(25,28)
>>> from sage.all import * >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(3))) >>> W.distinguished_reflection(Integer(1)) (1,4)(2,3)(5,6) >>> W.distinguished_reflection(Integer(2)) (1,3)(2,5)(4,6) >>> W.distinguished_reflection(Integer(3)) (1,5)(2,4)(3,6) >>> W = ReflectionGroup((Integer(3),Integer(1),Integer(1)),hyperplane_index_set=['a']) >>> W.distinguished_reflection('a') (1,2,3) >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(3)),(Integer(3),Integer(1),Integer(2))) >>> for i in range(W.number_of_reflection_hyperplanes()): ... W.distinguished_reflection(i+Integer(1)) (1,6)(2,5)(7,8) (1,5)(2,7)(6,8) (3,9,15)(4,10,16)(12,17,23)(14,18,24)(20,25,29)(21,22,26)(27,28,30) (3,11)(4,12)(9,13)(10,14)(15,19)(16,20)(17,21)(18,22)(23,27)(24,28)(25,26)(29,30) (1,7)(2,6)(5,8) (3,19)(4,25)(9,11)(10,17)(12,28)(13,15)(14,30)(16,18)(20,27)(21,29)(22,23)(24,26) (4,21,27)(10,22,28)(11,13,19)(12,14,20)(16,26,30)(17,18,25)(23,24,29) (3,13)(4,24)(9,19)(10,29)(11,15)(12,26)(14,21)(16,23)(17,30)(18,27)(20,22)(25,28)
- distinguished_reflections()[source]¶
Return a finite family containing the distinguished reflections of
selfindexed byhyperplane_index_set().These are the reflections in
selfacting on the complement of the fixed hyperplane \(H\) as \(\operatorname{exp}(2 \pi i / n)\), where \(n\) is the order of the reflection subgroup fixing \(H\).EXAMPLES:
sage: W = ReflectionGroup((1,1,3)) sage: W.distinguished_reflections() Finite family {1: (1,4)(2,3)(5,6), 2: (1,3)(2,5)(4,6), 3: (1,5)(2,4)(3,6)} sage: W = ReflectionGroup((1,1,3),hyperplane_index_set=['a','b','c']) sage: W.distinguished_reflections() Finite family {'a': (1,4)(2,3)(5,6), 'b': (1,3)(2,5)(4,6), 'c': (1,5)(2,4)(3,6)} sage: W = ReflectionGroup((3,1,1)) sage: W.distinguished_reflections() Finite family {1: (1,2,3)} sage: W = ReflectionGroup((1,1,3),(3,1,2)) sage: W.distinguished_reflections() Finite family {1: (1,6)(2,5)(7,8), 2: (1,5)(2,7)(6,8), 3: (3,9,15)(4,10,16)(12,17,23)(14,18,24)(20,25,29)(21,22,26)(27,28,30), 4: (3,11)(4,12)(9,13)(10,14)(15,19)(16,20)(17,21)(18,22)(23,27)(24,28)(25,26)(29,30), 5: (1,7)(2,6)(5,8), 6: (3,19)(4,25)(9,11)(10,17)(12,28)(13,15)(14,30)(16,18)(20,27)(21,29)(22,23)(24,26), 7: (4,21,27)(10,22,28)(11,13,19)(12,14,20)(16,26,30)(17,18,25)(23,24,29), 8: (3,13)(4,24)(9,19)(10,29)(11,15)(12,26)(14,21)(16,23)(17,30)(18,27)(20,22)(25,28)}
>>> from sage.all import * >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(3))) >>> W.distinguished_reflections() Finite family {1: (1,4)(2,3)(5,6), 2: (1,3)(2,5)(4,6), 3: (1,5)(2,4)(3,6)} >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(3)),hyperplane_index_set=['a','b','c']) >>> W.distinguished_reflections() Finite family {'a': (1,4)(2,3)(5,6), 'b': (1,3)(2,5)(4,6), 'c': (1,5)(2,4)(3,6)} >>> W = ReflectionGroup((Integer(3),Integer(1),Integer(1))) >>> W.distinguished_reflections() Finite family {1: (1,2,3)} >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(3)),(Integer(3),Integer(1),Integer(2))) >>> W.distinguished_reflections() Finite family {1: (1,6)(2,5)(7,8), 2: (1,5)(2,7)(6,8), 3: (3,9,15)(4,10,16)(12,17,23)(14,18,24)(20,25,29)(21,22,26)(27,28,30), 4: (3,11)(4,12)(9,13)(10,14)(15,19)(16,20)(17,21)(18,22)(23,27)(24,28)(25,26)(29,30), 5: (1,7)(2,6)(5,8), 6: (3,19)(4,25)(9,11)(10,17)(12,28)(13,15)(14,30)(16,18)(20,27)(21,29)(22,23)(24,26), 7: (4,21,27)(10,22,28)(11,13,19)(12,14,20)(16,26,30)(17,18,25)(23,24,29), 8: (3,13)(4,24)(9,19)(10,29)(11,15)(12,26)(14,21)(16,23)(17,30)(18,27)(20,22)(25,28)}
- fake_degrees()[source]¶
Return the list of the fake degrees associated to
self.The fake degrees are \(q\)-versions of the degree of the character. In particular, they sum to Hilbert series of the coinvariant algebra of
self.Note
The ordering follows the one in Chevie and is not compatible with the current implementation of
irredubile_characters().EXAMPLES:
sage: W = ReflectionGroup(12) sage: W.fake_degrees() [1, q^12, q^11 + q, q^8 + q^4, q^7 + q^5, q^6 + q^4 + q^2, q^10 + q^8 + q^6, q^9 + q^7 + q^5 + q^3] sage: W = ReflectionGroup(["H",4]) sage: W.cardinality() 14400 sage: sum(fdeg.subs(q=1)**2 for fdeg in W.fake_degrees()) 14400
>>> from sage.all import * >>> W = ReflectionGroup(Integer(12)) >>> W.fake_degrees() [1, q^12, q^11 + q, q^8 + q^4, q^7 + q^5, q^6 + q^4 + q^2, q^10 + q^8 + q^6, q^9 + q^7 + q^5 + q^3] >>> W = ReflectionGroup(["H",Integer(4)]) >>> W.cardinality() 14400 >>> sum(fdeg.subs(q=Integer(1))**Integer(2) for fdeg in W.fake_degrees()) 14400
- fundamental_invariants()[source]¶
Return the fundamental invariants of
self.EXAMPLES:
sage: W = ReflectionGroup((1,1,3)) sage: W.fundamental_invariants() (-2*x0^2 + 2*x0*x1 - 2*x1^2, 6*x0^2*x1 - 6*x0*x1^2) sage: W = ReflectionGroup((3,1,2)) sage: W.fundamental_invariants() (x0^3 + x1^3, x0^3*x1^3)
>>> from sage.all import * >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(3))) >>> W.fundamental_invariants() (-2*x0^2 + 2*x0*x1 - 2*x1^2, 6*x0^2*x1 - 6*x0*x1^2) >>> W = ReflectionGroup((Integer(3),Integer(1),Integer(2))) >>> W.fundamental_invariants() (x0^3 + x1^3, x0^3*x1^3)
- hyperplane_index_set()[source]¶
Return the index set of the hyperplanes of
self.EXAMPLES:
sage: W = ReflectionGroup((1,1,4)) sage: W.hyperplane_index_set() (1, 2, 3, 4, 5, 6) sage: W = ReflectionGroup((1,1,4), hyperplane_index_set=[1,3,'asdf',7,9,11]) sage: W.hyperplane_index_set() (1, 3, 'asdf', 7, 9, 11) sage: W = ReflectionGroup((1,1,4),hyperplane_index_set=('a','b','c','d','e','f')) sage: W.hyperplane_index_set() ('a', 'b', 'c', 'd', 'e', 'f')
>>> from sage.all import * >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(4))) >>> W.hyperplane_index_set() (1, 2, 3, 4, 5, 6) >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(4)), hyperplane_index_set=[Integer(1),Integer(3),'asdf',Integer(7),Integer(9),Integer(11)]) >>> W.hyperplane_index_set() (1, 3, 'asdf', 7, 9, 11) >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(4)),hyperplane_index_set=('a','b','c','d','e','f')) >>> W.hyperplane_index_set() ('a', 'b', 'c', 'd', 'e', 'f')
- independent_roots()[source]¶
Return a collection of simple roots generating the underlying vector space of
self.For well-generated groups, these are all simple roots. Otherwise, a linearly independent subset of the simple roots is chosen.
EXAMPLES:
sage: W = ReflectionGroup((1,1,3)) sage: W.independent_roots() Finite family {1: (1, 0), 2: (0, 1)} sage: W = ReflectionGroup((4,2,3)) sage: W.simple_roots() Finite family {1: (1, 0, 0), 2: (-E(4), 1, 0), 3: (-1, 1, 0), 4: (0, -1, 1)} sage: W.independent_roots() Finite family {1: (1, 0, 0), 2: (-E(4), 1, 0), 4: (0, -1, 1)}
>>> from sage.all import * >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(3))) >>> W.independent_roots() Finite family {1: (1, 0), 2: (0, 1)} >>> W = ReflectionGroup((Integer(4),Integer(2),Integer(3))) >>> W.simple_roots() Finite family {1: (1, 0, 0), 2: (-E(4), 1, 0), 3: (-1, 1, 0), 4: (0, -1, 1)} >>> W.independent_roots() Finite family {1: (1, 0, 0), 2: (-E(4), 1, 0), 4: (0, -1, 1)}
- index_set()[source]¶
Return the index set of the simple reflections of
self.EXAMPLES:
sage: W = ReflectionGroup((1,1,4)) sage: W.index_set() (1, 2, 3) sage: W = ReflectionGroup((1,1,4), index_set=[1,3,'asdf']) sage: W.index_set() (1, 3, 'asdf') sage: W = ReflectionGroup((1,1,4), index_set=('a', 'b', 'c')) sage: W.index_set() ('a', 'b', 'c')
>>> from sage.all import * >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(4))) >>> W.index_set() (1, 2, 3) >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(4)), index_set=[Integer(1),Integer(3),'asdf']) >>> W.index_set() (1, 3, 'asdf') >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(4)), index_set=('a', 'b', 'c')) >>> W.index_set() ('a', 'b', 'c')
- invariant_form(brute_force=False)[source]¶
Return the form that is invariant under the action of
self.This is unique only up to a global scalar on the irreducible components.
INPUT:
brute_force– ifTrue, the computation is done by applying the Reynolds operator; this is, the invariant form of \(e_i\) and \(e_j\) is computed as the sum \(\langle w(e_i), w(e_j)\rangle\), where \(\langle \cdot, \cdot\rangle\) is the standard scalar product
EXAMPLES:
sage: W = ReflectionGroup(['A',3]) sage: F = W.invariant_form(); F [ 1 -1/2 0] [-1/2 1 -1/2] [ 0 -1/2 1]
>>> from sage.all import * >>> W = ReflectionGroup(['A',Integer(3)]) >>> F = W.invariant_form(); F [ 1 -1/2 0] [-1/2 1 -1/2] [ 0 -1/2 1]
To check that this is indeed the invariant form, see:
sage: S = W.simple_reflections() sage: all( F == S[i].matrix()*F*S[i].matrix().transpose() for i in W.index_set() ) True sage: W = ReflectionGroup(['B',3]) sage: F = W.invariant_form(); F [ 1 -1 0] [-1 2 -1] [ 0 -1 2] sage: w = W.an_element().to_matrix() sage: w * F * w.transpose().conjugate() == F True sage: S = W.simple_reflections() sage: all( F == S[i].matrix()*F*S[i].matrix().transpose() for i in W.index_set() ) True sage: W = ReflectionGroup((3,1,2)) sage: F = W.invariant_form(); F [1 0] [0 1] sage: S = W.simple_reflections() sage: all( F == S[i].matrix()*F*S[i].matrix().transpose().conjugate() for i in W.index_set() ) True
>>> from sage.all import * >>> S = W.simple_reflections() >>> all( F == S[i].matrix()*F*S[i].matrix().transpose() for i in W.index_set() ) True >>> W = ReflectionGroup(['B',Integer(3)]) >>> F = W.invariant_form(); F [ 1 -1 0] [-1 2 -1] [ 0 -1 2] >>> w = W.an_element().to_matrix() >>> w * F * w.transpose().conjugate() == F True >>> S = W.simple_reflections() >>> all( F == S[i].matrix()*F*S[i].matrix().transpose() for i in W.index_set() ) True >>> W = ReflectionGroup((Integer(3),Integer(1),Integer(2))) >>> F = W.invariant_form(); F [1 0] [0 1] >>> S = W.simple_reflections() >>> all( F == S[i].matrix()*F*S[i].matrix().transpose().conjugate() for i in W.index_set() ) True
It also worked for badly generated groups:
sage: W = ReflectionGroup(7) sage: W.is_well_generated() False sage: F = W.invariant_form(); F [1 0] [0 1] sage: S = W.simple_reflections() sage: all( F == S[i].matrix()*F*S[i].matrix().transpose().conjugate() for i in W.index_set() ) True
>>> from sage.all import * >>> W = ReflectionGroup(Integer(7)) >>> W.is_well_generated() False >>> F = W.invariant_form(); F [1 0] [0 1] >>> S = W.simple_reflections() >>> all( F == S[i].matrix()*F*S[i].matrix().transpose().conjugate() for i in W.index_set() ) True
And also for reducible types:
sage: W = ReflectionGroup(['B',3],(4,2,3),4,7); W Reducible complex reflection group of rank 10 and type B3 x G(4,2,3) x ST4 x ST7 sage: F = W.invariant_form(); S = W.simple_reflections() sage: all( F == S[i].matrix()*F*S[i].matrix().transpose().conjugate() for i in W.index_set() ) True
>>> from sage.all import * >>> W = ReflectionGroup(['B',Integer(3)],(Integer(4),Integer(2),Integer(3)),Integer(4),Integer(7)); W Reducible complex reflection group of rank 10 and type B3 x G(4,2,3) x ST4 x ST7 >>> F = W.invariant_form(); S = W.simple_reflections() >>> all( F == S[i].matrix()*F*S[i].matrix().transpose().conjugate() for i in W.index_set() ) True
- invariant_form_standardization()[source]¶
Return the transformation of the space that turns the invariant form of
selfinto the standard scalar product.Let \(I\) be the invariant form of a complex reflection group, and let \(A\) be the Hermitian matrix such that \(A^2 = I\). The matrix \(A\) defines a change of basis such that the identity matrix is the invariant form. Indeed, we have
\[(A^{-1} x A) \mathcal{I} (A^{-1} y A)^* = A^{-1} x I y^* A^{-1} = A^{-1} I A^{-1} = \mathcal{I},\]where \(\mathcal{I}\) is the identity matrix.
EXAMPLES:
sage: W = ReflectionGroup((4,2,5)) sage: I = W.invariant_form() sage: A = W.invariant_form_standardization() sage: A^2 == I True
>>> from sage.all import * >>> W = ReflectionGroup((Integer(4),Integer(2),Integer(5))) >>> I = W.invariant_form() >>> A = W.invariant_form_standardization() >>> A**Integer(2) == I True
- irreducible_components()[source]¶
Return a list containing the irreducible components of
selfas finite reflection groups.EXAMPLES:
sage: W = ReflectionGroup((1,1,3)) sage: W.irreducible_components() [Irreducible real reflection group of rank 2 and type A2] sage: W = ReflectionGroup((1,1,3),(2,1,3)) sage: W.irreducible_components() [Irreducible real reflection group of rank 2 and type A2, Irreducible real reflection group of rank 3 and type B3]
>>> from sage.all import * >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(3))) >>> W.irreducible_components() [Irreducible real reflection group of rank 2 and type A2] >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(3)),(Integer(2),Integer(1),Integer(3))) >>> W.irreducible_components() [Irreducible real reflection group of rank 2 and type A2, Irreducible real reflection group of rank 3 and type B3]
- is_crystallographic()[source]¶
Return
Trueifselfis crystallographic.This is, if the field of definition is the rational field.
Todo
Make this more robust and do not use the matrix representation of the simple reflections.
EXAMPLES:
sage: W = ReflectionGroup((1,1,3)); W Irreducible real reflection group of rank 2 and type A2 sage: W.is_crystallographic() True sage: W = ReflectionGroup((2,1,3)); W Irreducible real reflection group of rank 3 and type B3 sage: W.is_crystallographic() True sage: W = ReflectionGroup(23); W Irreducible real reflection group of rank 3 and type H3 sage: W.is_crystallographic() False sage: W = ReflectionGroup((3,1,3)); W Irreducible complex reflection group of rank 3 and type G(3,1,3) sage: W.is_crystallographic() False sage: W = ReflectionGroup((4,2,2)); W Irreducible complex reflection group of rank 2 and type G(4,2,2) sage: W.is_crystallographic() False
>>> from sage.all import * >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(3))); W Irreducible real reflection group of rank 2 and type A2 >>> W.is_crystallographic() True >>> W = ReflectionGroup((Integer(2),Integer(1),Integer(3))); W Irreducible real reflection group of rank 3 and type B3 >>> W.is_crystallographic() True >>> W = ReflectionGroup(Integer(23)); W Irreducible real reflection group of rank 3 and type H3 >>> W.is_crystallographic() False >>> W = ReflectionGroup((Integer(3),Integer(1),Integer(3))); W Irreducible complex reflection group of rank 3 and type G(3,1,3) >>> W.is_crystallographic() False >>> W = ReflectionGroup((Integer(4),Integer(2),Integer(2))); W Irreducible complex reflection group of rank 2 and type G(4,2,2) >>> W.is_crystallographic() False
- iteration_tracking_words()[source]¶
Return an iterator going through all elements in
selfthat tracks the reduced expressions.This can be much slower than using the iteration as a permutation group with strong generating set.
EXAMPLES:
sage: W = ReflectionGroup((1,1,3)) sage: for w in W.iteration_tracking_words(): w () (1,4)(2,3)(5,6) (1,3)(2,5)(4,6) (1,6,2)(3,5,4) (1,2,6)(3,4,5) (1,5)(2,4)(3,6)
>>> from sage.all import * >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(3))) >>> for w in W.iteration_tracking_words(): w () (1,4)(2,3)(5,6) (1,3)(2,5)(4,6) (1,6,2)(3,5,4) (1,2,6)(3,4,5) (1,5)(2,4)(3,6)
- jacobian_of_fundamental_invariants(invs=None)[source]¶
Return the matrix \([ \partial_{x_i} F_j ]\), where
invsare are any polynomials \(F_1,\ldots,F_n\) in \(x_1,\ldots,x_n\).INPUT:
invs– (default: the fundamental invariants) the polynomials \(F_1, \ldots, F_n\)
EXAMPLES:
sage: W = ReflectionGroup(['A',2]) sage: W.fundamental_invariants() (-2*x0^2 + 2*x0*x1 - 2*x1^2, 6*x0^2*x1 - 6*x0*x1^2) sage: W.jacobian_of_fundamental_invariants() [ -4*x0 + 2*x1 2*x0 - 4*x1] [12*x0*x1 - 6*x1^2 6*x0^2 - 12*x0*x1]
>>> from sage.all import * >>> W = ReflectionGroup(['A',Integer(2)]) >>> W.fundamental_invariants() (-2*x0^2 + 2*x0*x1 - 2*x1^2, 6*x0^2*x1 - 6*x0*x1^2) >>> W.jacobian_of_fundamental_invariants() [ -4*x0 + 2*x1 2*x0 - 4*x1] [12*x0*x1 - 6*x1^2 6*x0^2 - 12*x0*x1]
- number_of_irreducible_components()[source]¶
Return the number of irreducible components of
self.EXAMPLES:
sage: W = ReflectionGroup((1,1,3)) sage: W.number_of_irreducible_components() 1 sage: W = ReflectionGroup((1,1,3),(2,1,3)) sage: W.number_of_irreducible_components() 2
>>> from sage.all import * >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(3))) >>> W.number_of_irreducible_components() 1 >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(3)),(Integer(2),Integer(1),Integer(3))) >>> W.number_of_irreducible_components() 2
- primitive_vector_field(invs=None)[source]¶
Return the primitive vector field of
selfis irreducible and well-generated.The primitive vector field is given as the coefficients (being rational functions) in the basis \(\partial_{x_1}, \ldots, \partial_{x_n}\).
This is the partial derivation along the unique invariant of degree given by the Coxeter number. It can be computed as the row of the inverse of the Jacobian given by the highest degree.
EXAMPLES:
sage: W = ReflectionGroup(['A',2]) sage: W.primitive_vector_field() (3*x1/(6*x0^2 - 6*x0*x1 - 12*x1^2), 1/(6*x0^2 - 6*x0*x1 - 12*x1^2))
>>> from sage.all import * >>> W = ReflectionGroup(['A',Integer(2)]) >>> W.primitive_vector_field() (3*x1/(6*x0^2 - 6*x0*x1 - 12*x1^2), 1/(6*x0^2 - 6*x0*x1 - 12*x1^2))
- rank()[source]¶
Return the rank of
self.This is the dimension of the underlying vector space.
EXAMPLES:
sage: W = ReflectionGroup((1,1,3)) sage: W.rank() 2 sage: W = ReflectionGroup((2,1,3)) sage: W.rank() 3 sage: W = ReflectionGroup((4,1,3)) sage: W.rank() 3 sage: W = ReflectionGroup((4,2,3)) sage: W.rank() 3
>>> from sage.all import * >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(3))) >>> W.rank() 2 >>> W = ReflectionGroup((Integer(2),Integer(1),Integer(3))) >>> W.rank() 3 >>> W = ReflectionGroup((Integer(4),Integer(1),Integer(3))) >>> W.rank() 3 >>> W = ReflectionGroup((Integer(4),Integer(2),Integer(3))) >>> W.rank() 3
- reflection(i)[source]¶
Return the
i-th reflection ofself.EXAMPLES:
sage: W = ReflectionGroup((1,1,3)) sage: W.reflection(1) (1,4)(2,3)(5,6) sage: W.reflection(2) (1,3)(2,5)(4,6) sage: W.reflection(3) (1,5)(2,4)(3,6) sage: W = ReflectionGroup((3,1,1),reflection_index_set=['a','b']) sage: W.reflection('a') (1,2,3) sage: W.reflection('b') (1,3,2)
>>> from sage.all import * >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(3))) >>> W.reflection(Integer(1)) (1,4)(2,3)(5,6) >>> W.reflection(Integer(2)) (1,3)(2,5)(4,6) >>> W.reflection(Integer(3)) (1,5)(2,4)(3,6) >>> W = ReflectionGroup((Integer(3),Integer(1),Integer(1)),reflection_index_set=['a','b']) >>> W.reflection('a') (1,2,3) >>> W.reflection('b') (1,3,2)
- reflection_character()[source]¶
Return the reflection characters of
self.EXAMPLES:
sage: W = ReflectionGroup((1,1,3)) sage: W.reflection_character() [2, 0, -1]
>>> from sage.all import * >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(3))) >>> W.reflection_character() [2, 0, -1]
- reflection_eigenvalues(w, is_class_representative=False)[source]¶
Return the reflection eigenvalue of
winself.INPUT:
is_class_representative– boolean (default:True) whether to compute instead on the conjugacy class representative
See also
EXAMPLES:
sage: W = ReflectionGroup((1,1,3)) sage: for w in W: ....: print('%s %s'%(w.reduced_word(), W.reflection_eigenvalues(w))) [] [0, 0] [2] [1/2, 0] [1] [1/2, 0] [1, 2] [1/3, 2/3] [2, 1] [1/3, 2/3] [1, 2, 1] [1/2, 0]
>>> from sage.all import * >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(3))) >>> for w in W: ... print('%s %s'%(w.reduced_word(), W.reflection_eigenvalues(w))) [] [0, 0] [2] [1/2, 0] [1] [1/2, 0] [1, 2] [1/3, 2/3] [2, 1] [1/3, 2/3] [1, 2, 1] [1/2, 0]
- reflection_eigenvalues_family()[source]¶
Return the reflection eigenvalues of
selfas a finite family indexed by the class representatives ofself.OUTPUT: list with entries \(k/n\) representing the eigenvalue \(\zeta_n^k\)
EXAMPLES:
sage: W = ReflectionGroup((1,1,3)) sage: W.reflection_eigenvalues_family() Finite family {(): [0, 0], (1,4)(2,3)(5,6): [1/2, 0], (1,6,2)(3,5,4): [1/3, 2/3]} sage: W = ReflectionGroup((3,1,2)) sage: reflection_eigenvalues = W.reflection_eigenvalues_family() sage: for elt in sorted(reflection_eigenvalues.keys()): ....: print('%s %s'%(elt, reflection_eigenvalues[elt])) () [0, 0] (1,3,9)(2,4,10)(6,11,17)(8,12,18)(14,19,23)(15,16,20)(21,22,24) [1/3, 0] (1,3,9)(2,16,24)(4,20,21)(5,7,13)(6,12,23)(8,19,17)(10,15,22)(11,18,14) [1/3, 1/3] (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,20)(23,24) [1/2, 0] (1,7,3,13,9,5)(2,8,16,19,24,17)(4,14,20,11,21,18)(6,15,12,22,23,10) [1/6, 2/3] (1,9,3)(2,10,4)(6,17,11)(8,18,12)(14,23,19)(15,20,16)(21,24,22) [2/3, 0] (1,9,3)(2,20,22)(4,15,24)(5,7,13)(6,18,19)(8,23,11)(10,16,21)(12,14,17) [1/3, 2/3] (1,9,3)(2,24,16)(4,21,20)(5,13,7)(6,23,12)(8,17,19)(10,22,15)(11,14,18) [2/3, 2/3] (1,13,9,7,3,5)(2,14,24,18,16,11)(4,6,21,23,20,12)(8,22,17,15,19,10) [1/3, 5/6] sage: W = ReflectionGroup(23) sage: reflection_eigenvalues = W.reflection_eigenvalues_family() sage: for elt in sorted(reflection_eigenvalues.keys()): ....: print('%s %s'%(elt, reflection_eigenvalues[elt])) () [0, 0, 0] (1,8,4)(2,21,3)(5,10,11)(6,18,17)(7,9,12)(13,14,15)(16,23,19)(20,25,26)(22,24,27)(28,29,30) [1/3, 2/3, 0] (1,16)(2,5)(4,7)(6,9)(8,10)(11,13)(12,14)(17,20)(19,22)(21,24)(23,25)(26,28)(27,29) [1/2, 0, 0] (1,16)(2,9)(3,18)(4,10)(5,6)(7,8)(11,14)(12,13)(17,24)(19,25)(20,21)(22,23)(26,29)(27,28) [1/2, 1/2, 0] (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30) [1/2, 1/2, 1/2] (1,19,20,2,7)(3,6,11,13,9)(4,5,17,22,16)(8,12,15,14,10)(18,21,26,28,24)(23,27,30,29,25) [1/5, 4/5, 0] (1,20,7,19,2)(3,11,9,6,13)(4,17,16,5,22)(8,15,10,12,14)(18,26,24,21,28)(23,30,25,27,29) [2/5, 3/5, 0] (1,23,26,29,22,16,8,11,14,7)(2,10,4,9,18,17,25,19,24,3)(5,21,27,30,28,20,6,12,15,13) [1/10, 1/2, 9/10] (1,24,17,16,9,2)(3,12,13,18,27,28)(4,21,29,19,6,14)(5,25,26,20,10,11)(7,23,30,22,8,15) [1/6, 1/2, 5/6] (1,29,8,7,26,16,14,23,22,11)(2,9,25,3,4,17,24,10,18,19)(5,30,6,13,27,20,15,21,28,12) [3/10, 1/2, 7/10]
>>> from sage.all import * >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(3))) >>> W.reflection_eigenvalues_family() Finite family {(): [0, 0], (1,4)(2,3)(5,6): [1/2, 0], (1,6,2)(3,5,4): [1/3, 2/3]} >>> W = ReflectionGroup((Integer(3),Integer(1),Integer(2))) >>> reflection_eigenvalues = W.reflection_eigenvalues_family() >>> for elt in sorted(reflection_eigenvalues.keys()): ... print('%s %s'%(elt, reflection_eigenvalues[elt])) () [0, 0] (1,3,9)(2,4,10)(6,11,17)(8,12,18)(14,19,23)(15,16,20)(21,22,24) [1/3, 0] (1,3,9)(2,16,24)(4,20,21)(5,7,13)(6,12,23)(8,19,17)(10,15,22)(11,18,14) [1/3, 1/3] (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,20)(23,24) [1/2, 0] (1,7,3,13,9,5)(2,8,16,19,24,17)(4,14,20,11,21,18)(6,15,12,22,23,10) [1/6, 2/3] (1,9,3)(2,10,4)(6,17,11)(8,18,12)(14,23,19)(15,20,16)(21,24,22) [2/3, 0] (1,9,3)(2,20,22)(4,15,24)(5,7,13)(6,18,19)(8,23,11)(10,16,21)(12,14,17) [1/3, 2/3] (1,9,3)(2,24,16)(4,21,20)(5,13,7)(6,23,12)(8,17,19)(10,22,15)(11,14,18) [2/3, 2/3] (1,13,9,7,3,5)(2,14,24,18,16,11)(4,6,21,23,20,12)(8,22,17,15,19,10) [1/3, 5/6] >>> W = ReflectionGroup(Integer(23)) >>> reflection_eigenvalues = W.reflection_eigenvalues_family() >>> for elt in sorted(reflection_eigenvalues.keys()): ... print('%s %s'%(elt, reflection_eigenvalues[elt])) () [0, 0, 0] (1,8,4)(2,21,3)(5,10,11)(6,18,17)(7,9,12)(13,14,15)(16,23,19)(20,25,26)(22,24,27)(28,29,30) [1/3, 2/3, 0] (1,16)(2,5)(4,7)(6,9)(8,10)(11,13)(12,14)(17,20)(19,22)(21,24)(23,25)(26,28)(27,29) [1/2, 0, 0] (1,16)(2,9)(3,18)(4,10)(5,6)(7,8)(11,14)(12,13)(17,24)(19,25)(20,21)(22,23)(26,29)(27,28) [1/2, 1/2, 0] (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30) [1/2, 1/2, 1/2] (1,19,20,2,7)(3,6,11,13,9)(4,5,17,22,16)(8,12,15,14,10)(18,21,26,28,24)(23,27,30,29,25) [1/5, 4/5, 0] (1,20,7,19,2)(3,11,9,6,13)(4,17,16,5,22)(8,15,10,12,14)(18,26,24,21,28)(23,30,25,27,29) [2/5, 3/5, 0] (1,23,26,29,22,16,8,11,14,7)(2,10,4,9,18,17,25,19,24,3)(5,21,27,30,28,20,6,12,15,13) [1/10, 1/2, 9/10] (1,24,17,16,9,2)(3,12,13,18,27,28)(4,21,29,19,6,14)(5,25,26,20,10,11)(7,23,30,22,8,15) [1/6, 1/2, 5/6] (1,29,8,7,26,16,14,23,22,11)(2,9,25,3,4,17,24,10,18,19)(5,30,6,13,27,20,15,21,28,12) [3/10, 1/2, 7/10]
- reflection_hyperplane(i, as_linear_functional=False, with_order=False)[source]¶
Return the
i-th reflection hyperplane ofself.The
i-th reflection hyperplane corresponds to theidistinguished reflection.INPUT:
i– an index in the index setas_linear_functionals– boolean (default:False); whether to return the hyperplane or its linear functional in the basis dual to the given root basis
EXAMPLES:
sage: W = ReflectionGroup((2,1,2)) sage: W.reflection_hyperplane(3) Vector space of degree 2 and dimension 1 over Rational Field Basis matrix: [1 0]
>>> from sage.all import * >>> W = ReflectionGroup((Integer(2),Integer(1),Integer(2))) >>> W.reflection_hyperplane(Integer(3)) Vector space of degree 2 and dimension 1 over Rational Field Basis matrix: [1 0]
One can ask for the result as a linear form:
sage: W.reflection_hyperplane(3, True) (0, 1)
>>> from sage.all import * >>> W.reflection_hyperplane(Integer(3), True) (0, 1)
- reflection_hyperplanes(as_linear_functionals=False, with_order=False)[source]¶
Return the list of all reflection hyperplanes of
self, either as a codimension 1 space, or as its linear functional.INPUT:
as_linear_functionals– boolean (default:False); whether to return the hyperplane or its linear functional in the basis dual to the given root basis
EXAMPLES:
sage: W = ReflectionGroup((1,1,3)) sage: for H in W.reflection_hyperplanes(): H Vector space of degree 2 and dimension 1 over Rational Field Basis matrix: [1 2] Vector space of degree 2 and dimension 1 over Rational Field Basis matrix: [ 1 1/2] Vector space of degree 2 and dimension 1 over Rational Field Basis matrix: [ 1 -1] sage: for H in W.reflection_hyperplanes(as_linear_functionals=True): H (1, -1/2) (1, -2) (1, 1) sage: W = ReflectionGroup((2,1,2)) sage: for H in W.reflection_hyperplanes(): H Vector space of degree 2 and dimension 1 over Rational Field Basis matrix: [1 1] Vector space of degree 2 and dimension 1 over Rational Field Basis matrix: [ 1 1/2] Vector space of degree 2 and dimension 1 over Rational Field Basis matrix: [1 0] Vector space of degree 2 and dimension 1 over Rational Field Basis matrix: [0 1] sage: for H in W.reflection_hyperplanes(as_linear_functionals=True): H (1, -1) (1, -2) (0, 1) (1, 0) sage: for H in W.reflection_hyperplanes(as_linear_functionals=True, with_order=True): H ((1, -1), 2) ((1, -2), 2) ((0, 1), 2) ((1, 0), 2)
>>> from sage.all import * >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(3))) >>> for H in W.reflection_hyperplanes(): H Vector space of degree 2 and dimension 1 over Rational Field Basis matrix: [1 2] Vector space of degree 2 and dimension 1 over Rational Field Basis matrix: [ 1 1/2] Vector space of degree 2 and dimension 1 over Rational Field Basis matrix: [ 1 -1] >>> for H in W.reflection_hyperplanes(as_linear_functionals=True): H (1, -1/2) (1, -2) (1, 1) >>> W = ReflectionGroup((Integer(2),Integer(1),Integer(2))) >>> for H in W.reflection_hyperplanes(): H Vector space of degree 2 and dimension 1 over Rational Field Basis matrix: [1 1] Vector space of degree 2 and dimension 1 over Rational Field Basis matrix: [ 1 1/2] Vector space of degree 2 and dimension 1 over Rational Field Basis matrix: [1 0] Vector space of degree 2 and dimension 1 over Rational Field Basis matrix: [0 1] >>> for H in W.reflection_hyperplanes(as_linear_functionals=True): H (1, -1) (1, -2) (0, 1) (1, 0) >>> for H in W.reflection_hyperplanes(as_linear_functionals=True, with_order=True): H ((1, -1), 2) ((1, -2), 2) ((0, 1), 2) ((1, 0), 2)
- reflection_index_set()[source]¶
Return the index set of the reflections of
self.EXAMPLES:
sage: W = ReflectionGroup((1,1,4)) sage: W.reflection_index_set() (1, 2, 3, 4, 5, 6) sage: W = ReflectionGroup((1,1,4), reflection_index_set=[1,3,'asdf',7,9,11]) sage: W.reflection_index_set() (1, 3, 'asdf', 7, 9, 11) sage: W = ReflectionGroup((1,1,4), reflection_index_set=('a','b','c','d','e','f')) sage: W.reflection_index_set() ('a', 'b', 'c', 'd', 'e', 'f')
>>> from sage.all import * >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(4))) >>> W.reflection_index_set() (1, 2, 3, 4, 5, 6) >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(4)), reflection_index_set=[Integer(1),Integer(3),'asdf',Integer(7),Integer(9),Integer(11)]) >>> W.reflection_index_set() (1, 3, 'asdf', 7, 9, 11) >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(4)), reflection_index_set=('a','b','c','d','e','f')) >>> W.reflection_index_set() ('a', 'b', 'c', 'd', 'e', 'f')
- reflections()[source]¶
Return a finite family containing the reflections of
self, indexed byself.reflection_index_set().EXAMPLES:
sage: W = ReflectionGroup((1,1,3)) sage: W.reflections() Finite family {1: (1,4)(2,3)(5,6), 2: (1,3)(2,5)(4,6), 3: (1,5)(2,4)(3,6)} sage: W = ReflectionGroup((1,1,3),reflection_index_set=['a','b','c']) sage: W.reflections() Finite family {'a': (1,4)(2,3)(5,6), 'b': (1,3)(2,5)(4,6), 'c': (1,5)(2,4)(3,6)} sage: W = ReflectionGroup((3,1,1)) sage: W.reflections() Finite family {1: (1,2,3), 2: (1,3,2)} sage: W = ReflectionGroup((1,1,3),(3,1,2)) sage: W.reflections() Finite family {1: (1,6)(2,5)(7,8), 2: (1,5)(2,7)(6,8), 3: (3,9,15)(4,10,16)(12,17,23)(14,18,24)(20,25,29)(21,22,26)(27,28,30), 4: (3,11)(4,12)(9,13)(10,14)(15,19)(16,20)(17,21)(18,22)(23,27)(24,28)(25,26)(29,30), 5: (1,7)(2,6)(5,8), 6: (3,19)(4,25)(9,11)(10,17)(12,28)(13,15)(14,30)(16,18)(20,27)(21,29)(22,23)(24,26), 7: (4,21,27)(10,22,28)(11,13,19)(12,14,20)(16,26,30)(17,18,25)(23,24,29), 8: (3,13)(4,24)(9,19)(10,29)(11,15)(12,26)(14,21)(16,23)(17,30)(18,27)(20,22)(25,28), 9: (3,15,9)(4,16,10)(12,23,17)(14,24,18)(20,29,25)(21,26,22)(27,30,28), 10: (4,27,21)(10,28,22)(11,19,13)(12,20,14)(16,30,26)(17,25,18)(23,29,24)}
>>> from sage.all import * >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(3))) >>> W.reflections() Finite family {1: (1,4)(2,3)(5,6), 2: (1,3)(2,5)(4,6), 3: (1,5)(2,4)(3,6)} >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(3)),reflection_index_set=['a','b','c']) >>> W.reflections() Finite family {'a': (1,4)(2,3)(5,6), 'b': (1,3)(2,5)(4,6), 'c': (1,5)(2,4)(3,6)} >>> W = ReflectionGroup((Integer(3),Integer(1),Integer(1))) >>> W.reflections() Finite family {1: (1,2,3), 2: (1,3,2)} >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(3)),(Integer(3),Integer(1),Integer(2))) >>> W.reflections() Finite family {1: (1,6)(2,5)(7,8), 2: (1,5)(2,7)(6,8), 3: (3,9,15)(4,10,16)(12,17,23)(14,18,24)(20,25,29)(21,22,26)(27,28,30), 4: (3,11)(4,12)(9,13)(10,14)(15,19)(16,20)(17,21)(18,22)(23,27)(24,28)(25,26)(29,30), 5: (1,7)(2,6)(5,8), 6: (3,19)(4,25)(9,11)(10,17)(12,28)(13,15)(14,30)(16,18)(20,27)(21,29)(22,23)(24,26), 7: (4,21,27)(10,22,28)(11,13,19)(12,14,20)(16,26,30)(17,18,25)(23,24,29), 8: (3,13)(4,24)(9,19)(10,29)(11,15)(12,26)(14,21)(16,23)(17,30)(18,27)(20,22)(25,28), 9: (3,15,9)(4,16,10)(12,23,17)(14,24,18)(20,29,25)(21,26,22)(27,30,28), 10: (4,27,21)(10,28,22)(11,19,13)(12,20,14)(16,30,26)(17,25,18)(23,29,24)}
- roots()[source]¶
Return all roots corresponding to all reflections of
self.EXAMPLES:
sage: W = ReflectionGroup((1,1,3)) sage: W.roots() [(1, 0), (0, 1), (1, 1), (-1, 0), (0, -1), (-1, -1)] sage: W = ReflectionGroup((3,1,2)) sage: W.roots() [(1, 0), (-1, 1), (E(3), 0), (-E(3), 1), (0, 1), (1, -1), (0, E(3)), (1, -E(3)), (E(3)^2, 0), (-E(3)^2, 1), (E(3), -1), (E(3), -E(3)), (0, E(3)^2), (1, -E(3)^2), (-1, E(3)), (-E(3), E(3)), (E(3)^2, -1), (E(3)^2, -E(3)), (E(3), -E(3)^2), (-E(3)^2, E(3)), (-1, E(3)^2), (-E(3), E(3)^2), (E(3)^2, -E(3)^2), (-E(3)^2, E(3)^2)] sage: W = ReflectionGroup((4,2,2)) sage: W.roots() [(1, 0), (-E(4), 1), (-1, 1), (-1, 0), (E(4), 1), (1, 1), (0, -E(4)), (E(4), -1), (E(4), E(4)), (0, E(4)), (E(4), -E(4)), (0, 1), (1, -E(4)), (1, -1), (0, -1), (1, E(4)), (-E(4), 0), (-1, E(4)), (E(4), 0), (-E(4), E(4)), (-E(4), -1), (-E(4), -E(4)), (-1, -E(4)), (-1, -1)] sage: W = ReflectionGroup((1,1,4), (3,1,2)) sage: W.roots() [(1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, -1, 1), (1, 1, 0, 0, 0), (0, 1, 1, 0, 0), (1, 1, 1, 0, 0), (-1, 0, 0, 0, 0), (0, -1, 0, 0, 0), (0, 0, -1, 0, 0), (-1, -1, 0, 0, 0), (0, -1, -1, 0, 0), (-1, -1, -1, 0, 0), (0, 0, 0, E(3), 0), (0, 0, 0, -E(3), 1), (0, 0, 0, 0, 1), (0, 0, 0, 1, -1), (0, 0, 0, 0, E(3)), (0, 0, 0, 1, -E(3)), (0, 0, 0, E(3)^2, 0), (0, 0, 0, -E(3)^2, 1), (0, 0, 0, E(3), -1), (0, 0, 0, E(3), -E(3)), (0, 0, 0, 0, E(3)^2), (0, 0, 0, 1, -E(3)^2), (0, 0, 0, -1, E(3)), (0, 0, 0, -E(3), E(3)), (0, 0, 0, E(3)^2, -1), (0, 0, 0, E(3)^2, -E(3)), (0, 0, 0, E(3), -E(3)^2), (0, 0, 0, -E(3)^2, E(3)), (0, 0, 0, -1, E(3)^2), (0, 0, 0, -E(3), E(3)^2), (0, 0, 0, E(3)^2, -E(3)^2), (0, 0, 0, -E(3)^2, E(3)^2)]
>>> from sage.all import * >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(3))) >>> W.roots() [(1, 0), (0, 1), (1, 1), (-1, 0), (0, -1), (-1, -1)] >>> W = ReflectionGroup((Integer(3),Integer(1),Integer(2))) >>> W.roots() [(1, 0), (-1, 1), (E(3), 0), (-E(3), 1), (0, 1), (1, -1), (0, E(3)), (1, -E(3)), (E(3)^2, 0), (-E(3)^2, 1), (E(3), -1), (E(3), -E(3)), (0, E(3)^2), (1, -E(3)^2), (-1, E(3)), (-E(3), E(3)), (E(3)^2, -1), (E(3)^2, -E(3)), (E(3), -E(3)^2), (-E(3)^2, E(3)), (-1, E(3)^2), (-E(3), E(3)^2), (E(3)^2, -E(3)^2), (-E(3)^2, E(3)^2)] >>> W = ReflectionGroup((Integer(4),Integer(2),Integer(2))) >>> W.roots() [(1, 0), (-E(4), 1), (-1, 1), (-1, 0), (E(4), 1), (1, 1), (0, -E(4)), (E(4), -1), (E(4), E(4)), (0, E(4)), (E(4), -E(4)), (0, 1), (1, -E(4)), (1, -1), (0, -1), (1, E(4)), (-E(4), 0), (-1, E(4)), (E(4), 0), (-E(4), E(4)), (-E(4), -1), (-E(4), -E(4)), (-1, -E(4)), (-1, -1)] >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(4)), (Integer(3),Integer(1),Integer(2))) >>> W.roots() [(1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, -1, 1), (1, 1, 0, 0, 0), (0, 1, 1, 0, 0), (1, 1, 1, 0, 0), (-1, 0, 0, 0, 0), (0, -1, 0, 0, 0), (0, 0, -1, 0, 0), (-1, -1, 0, 0, 0), (0, -1, -1, 0, 0), (-1, -1, -1, 0, 0), (0, 0, 0, E(3), 0), (0, 0, 0, -E(3), 1), (0, 0, 0, 0, 1), (0, 0, 0, 1, -1), (0, 0, 0, 0, E(3)), (0, 0, 0, 1, -E(3)), (0, 0, 0, E(3)^2, 0), (0, 0, 0, -E(3)^2, 1), (0, 0, 0, E(3), -1), (0, 0, 0, E(3), -E(3)), (0, 0, 0, 0, E(3)^2), (0, 0, 0, 1, -E(3)^2), (0, 0, 0, -1, E(3)), (0, 0, 0, -E(3), E(3)), (0, 0, 0, E(3)^2, -1), (0, 0, 0, E(3)^2, -E(3)), (0, 0, 0, E(3), -E(3)^2), (0, 0, 0, -E(3)^2, E(3)), (0, 0, 0, -1, E(3)^2), (0, 0, 0, -E(3), E(3)^2), (0, 0, 0, E(3)^2, -E(3)^2), (0, 0, 0, -E(3)^2, E(3)^2)]
- series()[source]¶
Return the series of the classification type to which
selfbelongs.For real reflection groups, these are the Cartan-Killing classification types “A”,”B”,”C”,”D”,”E”,”F”,”G”,”H”,”I”, and for complx non-real reflection groups these are the Shephard-Todd classification type “ST”.
EXAMPLES:
sage: ReflectionGroup((1,1,3)).series() ['A'] sage: ReflectionGroup((3,1,3)).series() ['ST']
>>> from sage.all import * >>> ReflectionGroup((Integer(1),Integer(1),Integer(3))).series() ['A'] >>> ReflectionGroup((Integer(3),Integer(1),Integer(3))).series() ['ST']
- set_reflection_representation(refl_repr=None)[source]¶
Set the reflection representation of
self.INPUT:
refl_repr– dictionary representing the matrices of the generators ofselfwith keys given by the index set, orNoneto reset to the default reflection representation
EXAMPLES:
sage: W = ReflectionGroup((1,1,3)) sage: for w in W: w.to_matrix(); print("-----") [1 0] [0 1] ----- [ 1 1] [ 0 -1] ----- [-1 0] [ 1 1] ----- [-1 -1] [ 1 0] ----- [ 0 1] [-1 -1] ----- [ 0 -1] [-1 0] ----- sage: W.set_reflection_representation({1: matrix([[0,1,0],[1,0,0],[0,0,1]]), 2: matrix([[1,0,0],[0,0,1],[0,1,0]])}) sage: for w in W: w.to_matrix(); print("-----") [1 0 0] [0 1 0] [0 0 1] ----- [1 0 0] [0 0 1] [0 1 0] ----- [0 1 0] [1 0 0] [0 0 1] ----- [0 0 1] [1 0 0] [0 1 0] ----- [0 1 0] [0 0 1] [1 0 0] ----- [0 0 1] [0 1 0] [1 0 0] ----- sage: W.set_reflection_representation()
>>> from sage.all import * >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(3))) >>> for w in W: w.to_matrix(); print("-----") [1 0] [0 1] ----- [ 1 1] [ 0 -1] ----- [-1 0] [ 1 1] ----- [-1 -1] [ 1 0] ----- [ 0 1] [-1 -1] ----- [ 0 -1] [-1 0] ----- >>> W.set_reflection_representation({Integer(1): matrix([[Integer(0),Integer(1),Integer(0)],[Integer(1),Integer(0),Integer(0)],[Integer(0),Integer(0),Integer(1)]]), Integer(2): matrix([[Integer(1),Integer(0),Integer(0)],[Integer(0),Integer(0),Integer(1)],[Integer(0),Integer(1),Integer(0)]])}) >>> for w in W: w.to_matrix(); print("-----") [1 0 0] [0 1 0] [0 0 1] ----- [1 0 0] [0 0 1] [0 1 0] ----- [0 1 0] [1 0 0] [0 0 1] ----- [0 0 1] [1 0 0] [0 1 0] ----- [0 1 0] [0 0 1] [1 0 0] ----- [0 0 1] [0 1 0] [1 0 0] ----- >>> W.set_reflection_representation()
- simple_coroot(i)[source]¶
Return the simple root with index
i.EXAMPLES:
sage: W = ReflectionGroup(['A',3]) sage: W.simple_coroot(1) (2, -1, 0)
>>> from sage.all import * >>> W = ReflectionGroup(['A',Integer(3)]) >>> W.simple_coroot(Integer(1)) (2, -1, 0)
- simple_coroots()[source]¶
Return the simple coroots of
self.These are the coroots corresponding to the simple reflections.
EXAMPLES:
sage: W = ReflectionGroup((1,1,3)) sage: W.simple_coroots() Finite family {1: (2, -1), 2: (-1, 2)} sage: W = ReflectionGroup((1,1,4), (2,1,2)) sage: W.simple_coroots() Finite family {1: (2, -1, 0, 0, 0), 2: (-1, 2, -1, 0, 0), 3: (0, -1, 2, 0, 0), 4: (0, 0, 0, 2, -2), 5: (0, 0, 0, -1, 2)} sage: W = ReflectionGroup((3,1,2)) sage: W.simple_coroots() Finite family {1: (-2*E(3) - E(3)^2, 0), 2: (-1, 1)} sage: W = ReflectionGroup((1,1,4), (3,1,2)) sage: W.simple_coroots() Finite family {1: (2, -1, 0, 0, 0), 2: (-1, 2, -1, 0, 0), 3: (0, -1, 2, 0, 0), 4: (0, 0, 0, -2*E(3) - E(3)^2, 0), 5: (0, 0, 0, -1, 1)}
>>> from sage.all import * >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(3))) >>> W.simple_coroots() Finite family {1: (2, -1), 2: (-1, 2)} >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(4)), (Integer(2),Integer(1),Integer(2))) >>> W.simple_coroots() Finite family {1: (2, -1, 0, 0, 0), 2: (-1, 2, -1, 0, 0), 3: (0, -1, 2, 0, 0), 4: (0, 0, 0, 2, -2), 5: (0, 0, 0, -1, 2)} >>> W = ReflectionGroup((Integer(3),Integer(1),Integer(2))) >>> W.simple_coroots() Finite family {1: (-2*E(3) - E(3)^2, 0), 2: (-1, 1)} >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(4)), (Integer(3),Integer(1),Integer(2))) >>> W.simple_coroots() Finite family {1: (2, -1, 0, 0, 0), 2: (-1, 2, -1, 0, 0), 3: (0, -1, 2, 0, 0), 4: (0, 0, 0, -2*E(3) - E(3)^2, 0), 5: (0, 0, 0, -1, 1)}
- simple_reflection(i)[source]¶
Return the
i-th simple reflection ofself.EXAMPLES:
sage: W = ReflectionGroup((1,1,3)) sage: W.simple_reflection(1) (1,4)(2,3)(5,6) sage: W.simple_reflections() Finite family {1: (1,4)(2,3)(5,6), 2: (1,3)(2,5)(4,6)}
>>> from sage.all import * >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(3))) >>> W.simple_reflection(Integer(1)) (1,4)(2,3)(5,6) >>> W.simple_reflections() Finite family {1: (1,4)(2,3)(5,6), 2: (1,3)(2,5)(4,6)}
- simple_root(i)[source]¶
Return the simple root with index
i.EXAMPLES:
sage: W = ReflectionGroup(['A',3]) sage: W.simple_root(1) (1, 0, 0) sage: W.simple_root(2) (0, 1, 0) sage: W.simple_root(3) (0, 0, 1)
>>> from sage.all import * >>> W = ReflectionGroup(['A',Integer(3)]) >>> W.simple_root(Integer(1)) (1, 0, 0) >>> W.simple_root(Integer(2)) (0, 1, 0) >>> W.simple_root(Integer(3)) (0, 0, 1)
- simple_roots()[source]¶
Return the simple roots of
self.These are the roots corresponding to the simple reflections.
EXAMPLES:
sage: W = ReflectionGroup((1,1,3)) sage: W.simple_roots() Finite family {1: (1, 0), 2: (0, 1)} sage: W = ReflectionGroup((1,1,4), (2,1,2)) sage: W.simple_roots() Finite family {1: (1, 0, 0, 0, 0), 2: (0, 1, 0, 0, 0), 3: (0, 0, 1, 0, 0), 4: (0, 0, 0, 1, 0), 5: (0, 0, 0, 0, 1)} sage: W = ReflectionGroup((3,1,2)) sage: W.simple_roots() Finite family {1: (1, 0), 2: (-1, 1)} sage: W = ReflectionGroup((1,1,4), (3,1,2)) sage: W.simple_roots() Finite family {1: (1, 0, 0, 0, 0), 2: (0, 1, 0, 0, 0), 3: (0, 0, 1, 0, 0), 4: (0, 0, 0, 1, 0), 5: (0, 0, 0, -1, 1)}
>>> from sage.all import * >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(3))) >>> W.simple_roots() Finite family {1: (1, 0), 2: (0, 1)} >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(4)), (Integer(2),Integer(1),Integer(2))) >>> W.simple_roots() Finite family {1: (1, 0, 0, 0, 0), 2: (0, 1, 0, 0, 0), 3: (0, 0, 1, 0, 0), 4: (0, 0, 0, 1, 0), 5: (0, 0, 0, 0, 1)} >>> W = ReflectionGroup((Integer(3),Integer(1),Integer(2))) >>> W.simple_roots() Finite family {1: (1, 0), 2: (-1, 1)} >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(4)), (Integer(3),Integer(1),Integer(2))) >>> W.simple_roots() Finite family {1: (1, 0, 0, 0, 0), 2: (0, 1, 0, 0, 0), 3: (0, 0, 1, 0, 0), 4: (0, 0, 0, 1, 0), 5: (0, 0, 0, -1, 1)}
- class sage.combinat.root_system.reflection_group_complex.IrreducibleComplexReflectionGroup(W_types, index_set=None, hyperplane_index_set=None, reflection_index_set=None)[source]¶
Bases:
ComplexReflectionGroup- class Element[source]¶
Bases:
Element- is_coxeter_element(which_primitive=1, is_class_representative=False)[source]¶
Return
Trueifselfis a Coxeter element.This is, whether
selfhas an eigenvalue that is a primitive \(h\)-th root of unity.INPUT:
which_primitive– (default:1) for which power of the first primitiveh-th root of unity to look as a reflection eigenvalue for a regular elementis_class_representative– boolean (default:True); whether to compute instead on the conjugacy class representative
See also
coxeter_element()coxeter_elements()EXAMPLES:
sage: W = ReflectionGroup((1,1,3)) sage: for w in W: ....: print('%s %s'%(w.reduced_word(), w.is_coxeter_element())) [] False [2] False [1] False [1, 2] True [2, 1] True [1, 2, 1] False
>>> from sage.all import * >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(3))) >>> for w in W: ... print('%s %s'%(w.reduced_word(), w.is_coxeter_element())) [] False [2] False [1] False [1, 2] True [2, 1] True [1, 2, 1] False
- is_h_regular(is_class_representative=False)[source]¶
Return whether
selfis regular.This is if
selfhas an eigenvector with eigenvalue \(h\) and which does not lie in any reflection hyperplane. Here, \(h\) denotes the Coxeter number.EXAMPLES:
sage: W = ReflectionGroup((1,1,3)) sage: for w in W: ....: print('%s %s'%(w.reduced_word(), w.is_h_regular())) [] False [2] False [1] False [1, 2] True [2, 1] True [1, 2, 1] False
>>> from sage.all import * >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(3))) >>> for w in W: ... print('%s %s'%(w.reduced_word(), w.is_h_regular())) [] False [2] False [1] False [1, 2] True [2, 1] True [1, 2, 1] False
- is_regular(h, is_class_representative=False)[source]¶
Return whether
selfis regular.This is, if
selfhas an eigenvector with eigenvalue of orderhand which does not lie in any reflection hyperplane.INPUT:
h– the order of the eigenvalueis_class_representative– boolean (default:True); whether to compute instead on the conjugacy class representative
EXAMPLES:
sage: W = ReflectionGroup((1,1,3)) sage: h = W.coxeter_number() sage: for w in W: ....: print("{} {}".format(w.reduced_word(), w.is_regular(h))) [] False [2] False [1] False [1, 2] True [2, 1] True [1, 2, 1] False sage: W = ReflectionGroup(23); h = W.coxeter_number() sage: for w in W: ....: if w.is_regular(h): ....: w.reduced_word() [1, 2, 3] [2, 1, 3] [1, 3, 2] [3, 2, 1] [2, 1, 2, 3, 2] [2, 3, 2, 1, 2] [1, 2, 1, 2, 3, 2, 1] [1, 2, 3, 2, 1, 2, 1] [1, 2, 1, 2, 3, 2, 1, 2, 3] [2, 1, 2, 1, 3, 2, 1, 2, 3] [2, 1, 2, 3, 2, 1, 2, 1, 3] [1, 2, 3, 2, 1, 2, 1, 3, 2] [3, 2, 1, 2, 1, 3, 2, 1, 2] [1, 2, 1, 2, 1, 3, 2, 1, 2] [2, 3, 2, 1, 2, 1, 3, 2, 1] [2, 1, 2, 1, 3, 2, 1, 2, 1] [2, 3, 2, 1, 2, 1, 3, 2, 1, 2, 3] [1, 3, 2, 1, 2, 1, 3, 2, 1, 2, 3] [1, 2, 1, 2, 1, 3, 2, 1, 2, 1, 3] [1, 2, 1, 2, 3, 2, 1, 2, 1, 3, 2] [1, 2, 3, 2, 1, 2, 1, 3, 2, 1, 2] [2, 1, 2, 3, 2, 1, 2, 1, 3, 2, 1] [2, 1, 2, 3, 2, 1, 2, 1, 3, 2, 1, 2, 3] [1, 2, 1, 3, 2, 1, 2, 1, 3, 2, 1, 2, 3]
>>> from sage.all import * >>> W = ReflectionGroup((Integer(1),Integer(1),Integer(3))) >>> h = W.coxeter_number() >>> for w in W: ... print("{} {}".format(w.reduced_word(), w.is_regular(h))) [] False [2] False [1] False [1, 2] True [2, 1] True [1, 2, 1] False >>> W = ReflectionGroup(Integer(23)); h = W.coxeter_number() >>> for w in W: ... if w.is_regular(h): ... w.reduced_word() [1, 2, 3] [2, 1, 3] [1, 3, 2] [3, 2, 1] [2, 1, 2, 3, 2] [2, 3, 2, 1, 2] [1, 2, 1, 2, 3, 2, 1] [1, 2, 3, 2, 1, 2, 1] [1, 2, 1, 2, 3, 2, 1, 2, 3] [2, 1, 2, 1, 3, 2, 1, 2, 3] [2, 1, 2, 3, 2, 1, 2, 1, 3] [1, 2, 3, 2, 1, 2, 1, 3, 2] [3, 2, 1, 2, 1, 3, 2, 1, 2] [1, 2, 1, 2, 1, 3, 2, 1, 2] [2, 3, 2, 1, 2, 1, 3, 2, 1] [2, 1, 2, 1, 3, 2, 1, 2, 1] [2, 3, 2, 1, 2, 1, 3, 2, 1, 2, 3] [1, 3, 2, 1, 2, 1, 3, 2, 1, 2, 3] [1, 2, 1, 2, 1, 3, 2, 1, 2, 1, 3] [1, 2, 1, 2, 3, 2, 1, 2, 1, 3, 2] [1, 2, 3, 2, 1, 2, 1, 3, 2, 1, 2] [2, 1, 2, 3, 2, 1, 2, 1, 3, 2, 1] [2, 1, 2, 3, 2, 1, 2, 1, 3, 2, 1, 2, 3] [1, 2, 1, 3, 2, 1, 2, 1, 3, 2, 1, 2, 3]
Check that Issue #25478 is fixed:
sage: W = ReflectionGroup(["A",5]) sage: w = W.from_reduced_word([1,2,3,5]) sage: w.is_regular(4) False sage: W = ReflectionGroup(["A",3]) sage: len([w for w in W if w.is_regular(w.order())]) 18
>>> from sage.all import * >>> W = ReflectionGroup(["A",Integer(5)]) >>> w = W.from_reduced_word([Integer(1),Integer(2),Integer(3),Integer(5)]) >>> w.is_regular(Integer(4)) False >>> W = ReflectionGroup(["A",Integer(3)]) >>> len([w for w in W if w.is_regular(w.order())]) 18
- sage.combinat.root_system.reflection_group_complex.multi_partitions(n, S, i=None)[source]¶
Return all vectors as lists of the same length as
Swhose standard inner product withSequalsn.EXAMPLES:
sage: from sage.combinat.root_system.reflection_group_complex import multi_partitions sage: multi_partitions(10, [2,3,3,4]) [[5, 0, 0, 0], [3, 0, 0, 1], [2, 2, 0, 0], [2, 1, 1, 0], [2, 0, 2, 0], [1, 0, 0, 2], [0, 2, 0, 1], [0, 1, 1, 1], [0, 0, 2, 1]]
>>> from sage.all import * >>> from sage.combinat.root_system.reflection_group_complex import multi_partitions >>> multi_partitions(Integer(10), [Integer(2),Integer(3),Integer(3),Integer(4)]) [[5, 0, 0, 0], [3, 0, 0, 1], [2, 2, 0, 0], [2, 1, 1, 0], [2, 0, 2, 0], [1, 0, 0, 2], [0, 2, 0, 1], [0, 1, 1, 1], [0, 0, 2, 1]]
- sage.combinat.root_system.reflection_group_complex.power(k)[source]¶
Return \(f^k\) and caching all intermediate results.
Speeds the computation if one has to compute \(f^k\)’s for many values of \(k\).
EXAMPLES:
sage: P.<x,y,z> = PolynomialRing(QQ) sage: f = -2*x^2 + 2*x*y - 2*y^2 + 2*y*z - 2*z^2 sage: all( f^k == power(f,k) for k in range(20) ) True
>>> from sage.all import * >>> P = PolynomialRing(QQ, names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3) >>> f = -Integer(2)*x**Integer(2) + Integer(2)*x*y - Integer(2)*y**Integer(2) + Integer(2)*y*z - Integer(2)*z**Integer(2) >>> all( f**k == power(f,k) for k in range(Integer(20)) ) True