# 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


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


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)


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


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


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)


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]


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]


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)#

A complex reflection group given as a permutation group.

class Element#

Bases: ComplexReflectionGroupElement

conjugacy_class()#

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)]

conjugacy_class_representative()#

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]

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))
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]

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])
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)

braid_relations()#

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]]]

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()
[ 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]

codegrees()#

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

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)

conjugacy_classes()#

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

conjugacy_classes_representatives()#

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]]

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])
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

degrees()#

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

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)


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)

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])
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

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])
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

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))
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)

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))
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)}

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)
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

fundamental_invariants()#

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)

hyperplane_index_set()#

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')

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))
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)}

index_set()#

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')

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])
sage: 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


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


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

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))
sage: I = W.invariant_form()
sage: A = W.invariant_form_standardization()
sage: A^2 == I
True

irreducible_components()#

Return a list containing the irreducible components of self as 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]

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
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

iteration_tracking_words()#

Return an iterator going through all elements in self that 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)

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])
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]

number_of_irreducible_components()#

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

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])
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))

rank()#

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

reflection(i)#

Return the i-th reflection of self.

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)

reflection_character()#

Return the reflection characters of self.

EXAMPLES:

sage: W = ReflectionGroup((1,1,3))
sage: W.reflection_character()
[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.

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]

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))
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]

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))
sage: W.reflection_hyperplane(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)

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))
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)

reflection_index_set()#

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')

reflections()#

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

roots()#

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)]

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()
['A']
sage: ReflectionGroup((3,1,3)).series()
['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))
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()

simple_coroot(i)#

Return the simple root with index i.

EXAMPLES:

sage: W = ReflectionGroup(['A',3])
sage: W.simple_coroot(1)
(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))
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)}

simple_reflection(i)#

Return the i-th simple reflection of self.

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)}

simple_root(i)#

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)

simple_roots()#

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)}

class sage.combinat.root_system.reflection_group_complex.IrreducibleComplexReflectionGroup(W_types, index_set=None, hyperplane_index_set=None, reflection_index_set=None)#
class Element#

Bases: 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

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

is_h_regular(is_class_representative=False)#

Return whether self is regular.

This is if self has 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

is_regular(h, is_class_representative=False)#

Return whether self is regular.

This is, if self has an eigenvector with eigenvalue of order h and which does not lie in any reflection hyperplane.

INPUT:

• h – the order of the eigenvalue

• is_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]


Check that github 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

sage.combinat.root_system.reflection_group_complex.multi_partitions(n, S, i=None)#

Return all vectors as lists of the same length as S whose standard inner product with S equals n.

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]]

sage.combinat.root_system.reflection_group_complex.power(k)#

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