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])                                      # optional - gap3
Irreducible real reflection group of rank 2 and type A2
sage: ReflectionGroup(['F',4])                                      # optional - gap3
Irreducible real reflection group of rank 4 and type F4
sage: ReflectionGroup(['H',3])                                      # optional - gap3
Irreducible real reflection group of rank 3 and type H3

or with Shephard-Todd types:

sage: ReflectionGroup((1,1,3))                                      # optional - gap3
Irreducible real reflection group of rank 2 and type A2
sage: ReflectionGroup((2,1,3))                                      # optional - gap3
Irreducible real reflection group of rank 3 and type B3
sage: ReflectionGroup((3,1,3))                                      # optional - gap3
Irreducible complex reflection group of rank 3 and type G(3,1,3)
sage: ReflectionGroup((4,2,3))                                      # optional - gap3
Irreducible complex reflection group of rank 3 and type G(4,2,3)
sage: ReflectionGroup(4)                                            # optional - gap3
Irreducible complex reflection group of rank 2 and type ST4
sage: ReflectionGroup(31)                                           # optional - gap3
Irreducible complex reflection group of rank 4 and type ST31

Also reducible types are allowed using concatenation:

sage: ReflectionGroup(['A',3],(4,2,3))                              # optional - gap3
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                               # optional - gap3
Reducible real reflection group of rank 2 and type A1 x A1
sage: W = ReflectionGroup((2,2,3)); W                               # optional - gap3
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                               # optional - gap3
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)                                          # optional - gap3
()
(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()                                  # optional - gap3
[]
[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()                   # optional - gap3
[]
[2]
[1]
[1, 2]
[1, 4]
[3]
[4]
[1, 3]

sage: for w in W: w.reduced_word(); w.to_matrix(); print("")        # optional - gap3
[]
[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]

The standard references for actions of complex reflection groups have the matrices acting on the right, so:

sage: W.simple_reflection(1).to_matrix()                            # optional - gap3
[-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_group
exponents 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.py
add a method reflection_to_root or distinguished_reflection_to_positive_root
diagrams 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_reflections into Gap4 or Sage
list 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)#

Bases: sage.structure.unique_representation.UniqueRepresentation, sage.groups.perm_gps.permgroup.PermutationGroup_generic

A complex reflection group given as a permutation group.

See also

ReflectionGroup()

class Element#

Bases: sage.combinat.root_system.reflection_group_element.ComplexReflectionGroupElement

conjugacy_class()#

Return the conjugacy class of self.

EXAMPLES:

sage: W = ReflectionGroup((1,1,3))                      # optional - gap3
sage: for w in W: sorted(w.conjugacy_class())           # optional - gap3
[()]
[(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()#

Return a representative of the conjugacy class of self.

EXAMPLES:

sage: W = ReflectionGroup((1,1,3))                      # optional - gap3
sage: for w in W:                                       # optional - gap3
....:     print('%s %s'%(w.reduced_word(), w.conjugacy_class_representative().reduced_word()))  # optional - gap3
[] []
[2] [1]
[1] [1]
[1, 2] [1, 2]
[2, 1] [1, 2]
[1, 2, 1] [1]

reflection_length(in_unitary_group=False)#

Return the reflection length of self.

This is the minimal numbers of reflections needed to obtain self.

INPUT:

in_unitary_group – (default: False) if True, the reflection length is computed in the unitary group which is the dimension of the move space of self

EXAMPLES:

sage: W = ReflectionGroup((1,1,3))                      # optional - gap3
sage: sorted([t.reflection_length() for t in W])        # optional - gap3
[0, 1, 1, 1, 2, 2]

sage: W = ReflectionGroup((2,1,2))                      # optional - gap3
sage: sorted([t.reflection_length() for t in W])        # optional - gap3
[0, 1, 1, 1, 1, 2, 2, 2]

sage: W = ReflectionGroup((2,2,2))                      # optional - gap3
sage: sorted([t.reflection_length() for t in W])        # optional - gap3
[0, 1, 1, 2]

sage: W = ReflectionGroup((3,1,2))                      # optional - gap3
sage: sorted([t.reflection_length() for t in W])        # optional - gap3
[0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]

apply_vector_field(f, vf=None)#

Returns a rational function obtained by applying the vector field vf to the rational function f.

If vf is not given, the primitive vector field is used.

EXAMPLES:

sage: W = ReflectionGroup(['A',2])               # optional - gap3
sage: for x in W.primitive_vector_field()[0].parent().gens():  # optional - gap3
....:     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()#

Return the braid relations of self.

EXAMPLES:

sage: W = ReflectionGroup((1,1,3))                          # optional - gap3
sage: W.braid_relations()                                   # optional - gap3
[[[1, 2, 1], [2, 1, 2]]]

sage: W = ReflectionGroup((2,1,3))                          # optional - gap3
sage: W.braid_relations()                                   # optional - gap3
[[[1, 2, 1, 2], [2, 1, 2, 1]], [[1, 3], [3, 1]], [[2, 3, 2], [3, 2, 3]]]

sage: W = ReflectionGroup((2,2,3))                          # optional - gap3
sage: W.braid_relations()                                   # optional - gap3
[[[1, 2, 1], [2, 1, 2]], [[1, 3], [3, 1]], [[2, 3, 2], [3, 2, 3]]]

cartan_matrix()#

Return the Cartan matrix associated with self.

If self is crystallographic, the returned Cartan matrix is an instance of CartanMatrix, and a normal matrix otherwise.

Let \(s_1, \ldots, s_n\) be a set of reflections which generate self with 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()              # optional - gap3
[ 2 -1  0  0]
[-1  2 -1  0]
[ 0 -1  2 -1]
[ 0  0 -1  2]

sage: ReflectionGroup(['H',4]).cartan_matrix()              # optional - gap3
[              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()                    # optional - gap3
[-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()              # optional - gap3
[       2  -2*E(4)       -2]
[    E(4)        2 1 - E(4)]
[      -1 1 + E(4)        2]

codegrees()#

Return the codegrees of self ordered within each irreducible component of self.

EXAMPLES:

sage: W = ReflectionGroup((1,1,4))                          # optional - gap3
sage: W.codegrees()                                         # optional - gap3
(2, 1, 0)

sage: W = ReflectionGroup((2,1,4))                          # optional - gap3
sage: W.codegrees()                                         # optional - gap3
(6, 4, 2, 0)

sage: W = ReflectionGroup((4,1,4))                          # optional - gap3
sage: W.codegrees()                                         # optional - gap3
(12, 8, 4, 0)

sage: W = ReflectionGroup((4,2,4))                          # optional - gap3
sage: W.codegrees()                                         # optional - gap3
(12, 8, 4, 0)

sage: W = ReflectionGroup((4,4,4))                          # optional - gap3
sage: W.codegrees()                                         # optional - gap3
(8, 8, 4, 0)

sage: W = ReflectionGroup((1,1,4), (3,1,2))                 # optional - gap3
sage: W.codegrees()                                         # optional - gap3
(2, 1, 0, 3, 0)

sage: W = ReflectionGroup((1,1,4), (6,1,12), 23)            # optional - gap3 # fails in GAP3
sage: W.codegrees()                                         # optional - gap3
(2, 1, 0, 66, 60, 54, 48, 42, 36, 30, 24, 18, 12, 6, 0, 8, 4, 0)

conjugacy_classes()#

Return the conjugacy classes of self.

EXAMPLES:

sage: W = ReflectionGroup((1,1,3))                          # optional - gap3
sage: for C in W.conjugacy_classes(): sorted(C)             # optional - gap3
[()]
[(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))                          # optional - gap3
sage: sum(len(C) for C in W.conjugacy_classes()) == W.cardinality() # optional - gap3
True

sage: W = ReflectionGroup((3,1,2))                          # optional - gap3
sage: sum(len(C) for C in W.conjugacy_classes()) == W.cardinality() # optional - gap3
True

sage: W = ReflectionGroup(23)                               # optional - gap3
sage: sum(len(C) for C in W.conjugacy_classes()) == W.cardinality() # optional - gap3
True

conjugacy_classes_representatives()#

Return the shortest representatives of the conjugacy classes of self.

EXAMPLES:

sage: W = ReflectionGroup((1,1,3))                          # optional - gap3
sage: [w.reduced_word() for w in W.conjugacy_classes_representatives()] # optional - gap3
[[], [1], [1, 2]]

sage: W = ReflectionGroup((1,1,4))                          # optional - gap3
sage: [w.reduced_word() for w in W.conjugacy_classes_representatives()] # optional - gap3
[[], [1], [1, 3], [1, 2], [1, 3, 2]]

sage: W = ReflectionGroup((3,1,2))                          # optional - gap3
sage: [w.reduced_word() for w in W.conjugacy_classes_representatives()] # optional - gap3
[[], [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)                               # optional - gap3
sage: [w.reduced_word() for w in W.conjugacy_classes_representatives()] # optional - gap3
    [[],
     [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)#

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])               # optional - gap3
sage: W.coxeter_number()                         # optional - gap3
30
sage: all(W.coxeter_number(chi).is_integer()    # optional - gap3
....:     for chi in W.irreducible_characters())
True
sage: W = ReflectionGroup(14)                    # optional - gap3
sage: W.coxeter_number()                         # optional - gap3
24

degrees()#

Return the degrees of self ordered within each irreducible component of self.

EXAMPLES:

sage: W = ReflectionGroup((1,1,4))                          # optional - gap3
sage: W.degrees()                                           # optional - gap3
(2, 3, 4)

sage: W = ReflectionGroup((2,1,4))                          # optional - gap3
sage: W.degrees()                                           # optional - gap3
(2, 4, 6, 8)

sage: W = ReflectionGroup((4,1,4))                          # optional - gap3
sage: W.degrees()                                           # optional - gap3
(4, 8, 12, 16)

sage: W = ReflectionGroup((4,2,4))                          # optional - gap3
sage: W.degrees()                                           # optional - gap3
(4, 8, 8, 12)

sage: W = ReflectionGroup((4,4,4))                          # optional - gap3
sage: W.degrees()                                           # optional - gap3
(4, 4, 8, 12)

Examples of reducible types:

sage: W = ReflectionGroup((1,1,4), (3,1,2)); W              # optional - gap3
Reducible complex reflection group of rank 5 and type A3 x G(3,1,2)
sage: W.degrees()                                           # optional - gap3
(2, 3, 4, 3, 6)

sage: W = ReflectionGroup((1,1,4), (6,1,12), 23)            # optional - gap3 # fails in GAP3
sage: W.degrees()                                           # optional - gap3
(2, 3, 4, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 2, 6, 10)

discriminant()#

Return the discriminant of self in 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])               # optional - gap3
sage: W.discriminant()                           # optional - gap3
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])               # optional - gap3
sage: W.discriminant()                           # optional - gap3
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)#

Return the discriminant of self in 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])               # optional - gap3
sage: W.discriminant_in_invariant_ring()         # optional - gap3
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])               # optional - gap3
sage: W.discriminant_in_invariant_ring()         # optional - gap3
-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])               # optional - gap3
sage: W.discriminant_in_invariant_ring()  # long time  # optional - gap3
(-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)#

Return the i-th distinguished reflection of self.

These are the reflections in self acting 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))                          # optional - gap3
sage: W.distinguished_reflection(1)                         # optional - gap3
(1,4)(2,3)(5,6)
sage: W.distinguished_reflection(2)                         # optional - gap3
(1,3)(2,5)(4,6)
sage: W.distinguished_reflection(3)                         # optional - gap3
(1,5)(2,4)(3,6)

sage: W = ReflectionGroup((3,1,1),hyperplane_index_set=['a'])   # optional - gap3
sage: W.distinguished_reflection('a')                       # optional - gap3
(1,2,3)

sage: W = ReflectionGroup((1,1,3),(3,1,2))                  # optional - gap3
sage: for i in range(W.number_of_reflection_hyperplanes()): # optional - gap3
....:     W.distinguished_reflection(i+1)                   # optional - gap3
(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()#

Return a finite family containing the distinguished reflections of self indexed by hyperplane_index_set().

These are the reflections in self acting 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))                          # optional - gap3
sage: W.distinguished_reflections()                         # optional - gap3
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'])   # optional - gap3
sage: W.distinguished_reflections()                         # optional - gap3
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))                          # optional - gap3
sage: W.distinguished_reflections()                         # optional - gap3
Finite family {1: (1,2,3)}

sage: W = ReflectionGroup((1,1,3),(3,1,2))                  # optional - gap3
sage: W.distinguished_reflections()                         # optional - gap3
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()#

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)                              # optional - gap3
sage: W.fake_degrees()                                     # optional - gap3
[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])                         # optional - gap3
sage: W.cardinality()                                      # optional - gap3
14400
sage: sum(fdeg.subs(q=1)**2 for fdeg in W.fake_degrees())  # optional - gap3
14400

fundamental_invariants()#

Return the fundamental invariants of self.

EXAMPLES:

sage: W = ReflectionGroup((1,1,3))                          # optional - gap3
sage: W.fundamental_invariants()                            # optional - gap3
(-2*x0^2 + 2*x0*x1 - 2*x1^2, 6*x0^2*x1 - 6*x0*x1^2)

sage: W = ReflectionGroup((3,1,2))                          # optional - gap3
sage: W.fundamental_invariants()                            # optional - gap3
(x0^3 + x1^3, x0^3*x1^3)

hyperplane_index_set()#

Return the index set of the hyperplanes of self.

EXAMPLES:

sage: W = ReflectionGroup((1,1,4))                          # optional - gap3
sage: W.hyperplane_index_set()                              # optional - gap3
(1, 2, 3, 4, 5, 6)
sage: W = ReflectionGroup((1,1,4), hyperplane_index_set=[1,3,'asdf',7,9,11])    # optional - gap3
sage: W.hyperplane_index_set()                              # optional - gap3
(1, 3, 'asdf', 7, 9, 11)
sage: W = ReflectionGroup((1,1,4),hyperplane_index_set=('a','b','c','d','e','f'))   # optional - gap3
sage: W.hyperplane_index_set()                              # optional - gap3
('a', 'b', 'c', 'd', 'e', 'f')

independent_roots()#

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))                          # optional - gap3
sage: W.independent_roots()                                 # optional - gap3
Finite family {1: (1, 0), 2: (0, 1)}

sage: W = ReflectionGroup((4,2,3))                          # optional - gap3
sage: W.simple_roots()                                      # optional - gap3
Finite family {1: (1, 0, 0), 2: (-E(4), 1, 0), 3: (-1, 1, 0), 4: (0, -1, 1)}
sage: W.independent_roots()                                 # optional - gap3
Finite family {1: (1, 0, 0), 2: (-E(4), 1, 0), 4: (0, -1, 1)}

index_set()#

Return the index set of the simple reflections of self.

EXAMPLES:

sage: W = ReflectionGroup((1,1,4))                          # optional - gap3
sage: W.index_set()                                         # optional - gap3
(1, 2, 3)
sage: W = ReflectionGroup((1,1,4), index_set=[1,3,'asdf'])  # optional - gap3
sage: W.index_set()                                         # optional - gap3
(1, 3, 'asdf')
sage: W = ReflectionGroup((1,1,4), index_set=('a', 'b', 'c'))   # optional - gap3
sage: W.index_set()                                         # optional - gap3
('a', 'b', 'c')

invariant_form(brute_force=False)#

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 – if True, 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])                          # optional - gap3
sage: F = W.invariant_form(); F                             # optional - gap3
[   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()                            # optional - gap3
sage: all( F == S[i].matrix()*F*S[i].matrix().transpose() for i in W.index_set() )  # optional - gap3
True

sage: W = ReflectionGroup(['B',3])                          # optional - gap3
sage: F = W.invariant_form(); F                             # optional - gap3
[ 1 -1  0]
[-1  2 -1]
[ 0 -1  2]
sage: w = W.an_element().to_matrix()                        # optional - gap3
sage: w * F * w.transpose().conjugate() == F                # optional - gap3
True

sage: S = W.simple_reflections()                            # optional - gap3
sage: all( F == S[i].matrix()*F*S[i].matrix().transpose() for i in W.index_set() )  # optional - gap3
True

sage: W = ReflectionGroup((3,1,2))                          # optional - gap3
sage: F = W.invariant_form(); F                             # optional - gap3
[1 0]
[0 1]

sage: S = W.simple_reflections()                            # optional - gap3
sage: all( F == S[i].matrix()*F*S[i].matrix().transpose().conjugate() for i in W.index_set() )  # optional - gap3
True

It also worked for badly generated groups:

sage: W = ReflectionGroup(7)                                # optional - gap3
sage: W.is_well_generated()                                 # optional - gap3
False

sage: F = W.invariant_form(); F                             # optional - gap3
[1 0]
[0 1]
sage: S = W.simple_reflections()                            # optional - gap3
sage: all( F == S[i].matrix()*F*S[i].matrix().transpose().conjugate() for i in W.index_set() )  # optional - gap3
True

And also for reducible types:

sage: W = ReflectionGroup(['B',3],(4,2,3),4,7); W           # optional - gap3
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()    # optional - gap3
sage: all( F == S[i].matrix()*F*S[i].matrix().transpose().conjugate() for i in W.index_set() )  # optional - gap3
True

invariant_form_standardization()#

Return the transformation of the space that turns the invariant form of self into 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))             # optional - gap3
sage: I = W.invariant_form()                   # optional - gap3
sage: A = W.invariant_form_standardization()   # optional - gap3
sage: A^2 == I                                 # optional - gap3
True

irreducible_components()#

Return a list containing the irreducible components of self as finite reflection groups.

EXAMPLES:

sage: W = ReflectionGroup((1,1,3))                          # optional - gap3
sage: W.irreducible_components()                            # optional - gap3
[Irreducible real reflection group of rank 2 and type A2]

sage: W = ReflectionGroup((1,1,3),(2,1,3))                  # optional - gap3
sage: W.irreducible_components()                            # optional - gap3
[Irreducible real reflection group of rank 2 and type A2,
Irreducible real reflection group of rank 3 and type B3]

is_crystallographic()#

Return True if self is 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                       # optional - gap3
Irreducible real reflection group of rank 2 and type A2
sage: W.is_crystallographic()                               # optional - gap3
True

sage: W = ReflectionGroup((2,1,3)); W                       # optional - gap3
Irreducible real reflection group of rank 3 and type B3
sage: W.is_crystallographic()                               # optional - gap3
True

sage: W = ReflectionGroup(23); W                            # optional - gap3
Irreducible real reflection group of rank 3 and type H3
sage: W.is_crystallographic()                               # optional - gap3
False

sage: W = ReflectionGroup((3,1,3)); W                       # optional - gap3
Irreducible complex reflection group of rank 3 and type G(3,1,3)
sage: W.is_crystallographic()                               # optional - gap3
False

sage: W = ReflectionGroup((4,2,2)); W                       # optional - gap3
Irreducible complex reflection group of rank 2 and type G(4,2,2)
sage: W.is_crystallographic()                               # optional - gap3
False

jacobian_of_fundamental_invariants(invs=None)#

Return the matrix \([ \partial_{x_i} F_j ]\), where invs are 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])               # optional - gap3
sage: W.fundamental_invariants()                 # optional - gap3
(-2*x0^2 + 2*x0*x1 - 2*x1^2, 6*x0^2*x1 - 6*x0*x1^2)

sage: W.jacobian_of_fundamental_invariants()     # optional - gap3
[     -4*x0 + 2*x1       2*x0 - 4*x1]
[12*x0*x1 - 6*x1^2 6*x0^2 - 12*x0*x1]

number_of_irreducible_components()#

Return the number of irreducible components of self.

EXAMPLES:

sage: W = ReflectionGroup((1,1,3))                          # optional - gap3
sage: W.number_of_irreducible_components()                  # optional - gap3
1

sage: W = ReflectionGroup((1,1,3),(2,1,3))                  # optional - gap3
sage: W.number_of_irreducible_components()                  # optional - gap3
2

primitive_vector_field(invs=None)#

Return the primitive vector field of self is 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])               # optional - gap3
sage: W.primitive_vector_field()                 # optional - gap3
(3*x1/(6*x0^2 - 6*x0*x1 - 12*x1^2), 1/(6*x0^2 - 6*x0*x1 - 12*x1^2))

rank()#

Return the rank of self.

This is the dimension of the underlying vector space.

EXAMPLES:

sage: W = ReflectionGroup((1,1,3))                          # optional - gap3
sage: W.rank()                                              # optional - gap3
2
sage: W = ReflectionGroup((2,1,3))                          # optional - gap3
sage: W.rank()                                              # optional - gap3
3
sage: W = ReflectionGroup((4,1,3))                          # optional - gap3
sage: W.rank()                                              # optional - gap3
3
sage: W = ReflectionGroup((4,2,3))                          # optional - gap3
sage: W.rank()                                              # optional - gap3
3

reflection(i)#

Return the i-th reflection of self.

EXAMPLES:

sage: W = ReflectionGroup((1,1,3))                          # optional - gap3
sage: W.reflection(1)                                       # optional - gap3
(1,4)(2,3)(5,6)
sage: W.reflection(2)                                       # optional - gap3
(1,3)(2,5)(4,6)
sage: W.reflection(3)                                       # optional - gap3
(1,5)(2,4)(3,6)

sage: W = ReflectionGroup((3,1,1),reflection_index_set=['a','b'])   # optional - gap3
sage: W.reflection('a')                                     # optional - gap3
(1,2,3)
sage: W.reflection('b')                                     # optional - gap3
(1,3,2)

reflection_character()#

Return the reflection characters of self.

EXAMPLES:

sage: W = ReflectionGroup((1,1,3))                          # optional - gap3
sage: W.reflection_character()                              # optional - gap3
[2, 0, -1]

reflection_eigenvalues(w, is_class_representative=False)#

Return the reflection eigenvalue of w in self.

INPUT:

is_class_representative – boolean (default True) whether to compute instead on the conjugacy class representative.

See also

reflection_eigenvalues_family()

EXAMPLES:

sage: W = ReflectionGroup((1,1,3))                          # optional - gap3
sage: for w in W:                                           # optional - gap3
....:     print('%s %s'%(w.reduced_word(), W.reflection_eigenvalues(w)))    # optional - gap3
[] [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()#

Return the reflection eigenvalues of self as a finite family indexed by the class representatives of self.

OUTPUT:

list with entries \(k/n\) representing the eigenvalue \(\zeta_n^k\).

EXAMPLES:

sage: W = ReflectionGroup((1,1,3))                          # optional - gap3
sage: W.reflection_eigenvalues_family()                     # optional - gap3
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))                          # optional - gap3
sage: reflection_eigenvalues = W.reflection_eigenvalues_family()    # optional - gap3
sage: for elt in sorted(reflection_eigenvalues.keys()):     # optional - gap3
....:     print('%s %s'%(elt, reflection_eigenvalues[elt])) # optional - gap3
() [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)                               # optional - gap3
sage: reflection_eigenvalues = W.reflection_eigenvalues_family()    # optional - gap3
sage: for elt in sorted(reflection_eigenvalues.keys()):     # optional - gap3
....:     print('%s %s'%(elt, reflection_eigenvalues[elt])) # optional - gap3
() [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)#

Return the i-th reflection hyperplane of self.

The i-th reflection hyperplane corresponds to the i distinguished reflection.

INPUT:

i – an index in the index set
as_linear_functionals – (default:False) flag 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))                          # optional - gap3
sage: W.reflection_hyperplane(3)                            # optional - gap3
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)                      # optional - gap3
(0, 1)

reflection_hyperplanes(as_linear_functionals=False, with_order=False)#

Return the list of all reflection hyperplanes of self, either as a codimension 1 space, or as its linear functional.

INPUT:

as_linear_functionals – (default:False) flag 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))                          # optional - gap3
sage: for H in W.reflection_hyperplanes(): H                # optional - gap3
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  # optional - gap3
(1, -1/2)
(1, -2)
(1, 1)


sage: W = ReflectionGroup((2,1,2))                          # optional - gap3
sage: for H in W.reflection_hyperplanes(): H                # optional - gap3
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  # optional - gap3
(1, -1)
(1, -2)
(0, 1)
(1, 0)

sage: for H in W.reflection_hyperplanes(as_linear_functionals=True, with_order=True): H  # optional - gap3
((1, -1), 2)
((1, -2), 2)
((0, 1), 2)
((1, 0), 2)

reflection_index_set()#

Return the index set of the reflections of self.

EXAMPLES:

sage: W = ReflectionGroup((1,1,4))                          # optional - gap3
sage: W.reflection_index_set()                              # optional - gap3
(1, 2, 3, 4, 5, 6)
sage: W = ReflectionGroup((1,1,4), reflection_index_set=[1,3,'asdf',7,9,11])    # optional - gap3
sage: W.reflection_index_set()                              # optional - gap3
(1, 3, 'asdf', 7, 9, 11)
sage: W = ReflectionGroup((1,1,4), reflection_index_set=('a','b','c','d','e','f'))  # optional - gap3
sage: W.reflection_index_set()                              # optional - gap3
('a', 'b', 'c', 'd', 'e', 'f')

reflections()#

Return a finite family containing the reflections of self, indexed by self.reflection_index_set().

EXAMPLES:

sage: W = ReflectionGroup((1,1,3))                          # optional - gap3
sage: W.reflections()                                       # optional - gap3
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'])   # optional - gap3
sage: W.reflections()                                       # optional - gap3
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))                          # optional - gap3
sage: W.reflections()                                       # optional - gap3
Finite family {1: (1,2,3), 2: (1,3,2)}

sage: W = ReflectionGroup((1,1,3),(3,1,2))                  # optional - gap3
sage: W.reflections()                                       # optional - gap3
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()#

Return all roots corresponding to all reflections of self.

EXAMPLES:

sage: W = ReflectionGroup((1,1,3))                          # optional - gap3
sage: W.roots()                                             # optional - gap3
[(1, 0), (0, 1), (1, 1), (-1, 0), (0, -1), (-1, -1)]

sage: W = ReflectionGroup((3,1,2))                          # optional - gap3
sage: W.roots()                                             # optional - gap3
[(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))                          # optional - gap3
sage: W.roots()                                             # optional - gap3
[(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))                 # optional - gap3
sage: W.roots()                                             # optional - gap3
[(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()#

Return the series of the classification type to which self belongs.

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()                     # optional - gap3
['A']
sage: ReflectionGroup((3,1,3)).series()                     # optional - gap3
['ST']

set_reflection_representation(refl_repr=None)#

Set the reflection representation of self.

INPUT:

refl_repr – a dictionary representing the matrices of the generators of self with keys given by the index set, or None to reset to the default reflection representation

EXAMPLES:

sage: W = ReflectionGroup((1,1,3))                          # optional - gap3
sage: for w in W: w.to_matrix(); print("-----")             # optional - gap3
[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]])}) # optional - gap3
sage: for w in W: w.to_matrix(); print("-----")             # optional - gap3
[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()                     # optional - gap3

simple_coroot(i)#

Return the simple root with index i.

EXAMPLES:

sage: W = ReflectionGroup(['A',3])                          # optional - gap3
sage: W.simple_coroot(1)                                    # optional - gap3
(2, -1, 0)

simple_coroots()#

Return the simple coroots of self.

These are the coroots corresponding to the simple reflections.

EXAMPLES:

sage: W = ReflectionGroup((1,1,3))                          # optional - gap3
sage: W.simple_coroots()                                    # optional - gap3
Finite family {1: (2, -1), 2: (-1, 2)}

sage: W = ReflectionGroup((1,1,4), (2,1,2))                 # optional - gap3
sage: W.simple_coroots()                                    # optional - gap3
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))                          # optional - gap3
sage: W.simple_coroots()                                    # optional - gap3
Finite family {1: (-2*E(3) - E(3)^2, 0), 2: (-1, 1)}

sage: W = ReflectionGroup((1,1,4), (3,1,2))                 # optional - gap3
sage: W.simple_coroots()                                    # optional - gap3
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)#

Return the i-th simple reflection of self.

EXAMPLES:

sage: W = ReflectionGroup((1,1,3))                          # optional - gap3
sage: W.simple_reflection(1)                                # optional - gap3
(1,4)(2,3)(5,6)
sage: W.simple_reflections()                                # optional - gap3
Finite family {1: (1,4)(2,3)(5,6), 2: (1,3)(2,5)(4,6)}

simple_root(i)#

Return the simple root with index i.

EXAMPLES:

sage: W = ReflectionGroup(['A',3])                          # optional - gap3
sage: W.simple_root(1)                                      # optional - gap3
(1, 0, 0)
sage: W.simple_root(2)                                      # optional - gap3
(0, 1, 0)
sage: W.simple_root(3)                                      # optional - gap3
(0, 0, 1)

simple_roots()#

Return the simple roots of self.

These are the roots corresponding to the simple reflections.

EXAMPLES:

sage: W = ReflectionGroup((1,1,3))                          # optional - gap3
sage: W.simple_roots()                                      # optional - gap3
Finite family {1: (1, 0), 2: (0, 1)}

sage: W = ReflectionGroup((1,1,4), (2,1,2))                 # optional - gap3
sage: W.simple_roots()                                      # optional - gap3
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))                          # optional - gap3
sage: W.simple_roots()                                      # optional - gap3
Finite family {1: (1, 0), 2: (-1, 1)}

sage: W = ReflectionGroup((1,1,4), (3,1,2))                 # optional - gap3
sage: W.simple_roots()                                      # optional - gap3
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)#

Bases: sage.combinat.root_system.reflection_group_complex.ComplexReflectionGroup

class Element#

Bases: sage.combinat.root_system.reflection_group_complex.ComplexReflectionGroup.Element

is_coxeter_element(which_primitive=1, is_class_representative=False)#

Return True if self is a Coxeter element.

This is, whether self has an eigenvalue that is a primitive \(h\)-th root of unity.

INPUT:

which_primitive – (default:1) for which power of the first primitive h-th root of unity to look as a reflection eigenvalue for a regular element
is_class_representative – boolean (default True) whether to compute instead on the conjugacy class representative