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
ordistinguished_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 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)#
Bases:
UniqueRepresentation
,PermutationGroup_generic
A complex reflection group given as a permutation group.
See also
- class Element#
Bases:
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())) [] [] [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
) 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)) # 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 functionf
.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 ofCartanMatrix
, 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 ofself
.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 ofself
.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 ofself
.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) (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 byhyperplane_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
– 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]) # 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
- 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)) # optional - gap3 sage: for w in W.iteration_tracking_words(): w # optional - gap3 () (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]) # 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 ofself
.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
inself
.INPUT:
is_class_representative
– boolean (defaultTrue
) whether to compute instead on the conjugacy class representative.
See also
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))) [] [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 ofself
.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])) () [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])) () [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 ofself
.The
i
-th reflection hyperplane corresponds to thei
distinguished reflection.INPUT:
i
– an index in the index setas_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 byself.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 ofself
with keys given by the index set, orNone
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 ofself
.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:
ComplexReflectionGroup
- class Element#
Bases:
Element
- is_coxeter_element(which_primitive=1, is_class_representative=False)#
Return
True
ifself
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 primitiveh
-th root of unity to look as a reflection eigenvalue for a regular elementis_class_representative
– boolean (defaultTrue
) whether to compute instead on the conjugacy class representative
See also
coxeter_element()
coxeter_elements()
EXAMPLES:
sage: W = ReflectionGroup((1,1,3)) # optional - gap3 sage: for w in W: # optional - gap3 ....: 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)) # optional - gap3 sage: for w in W: # optional - gap3 ....: 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 orderh
and which does not lie in any reflection hyperplane.INPUT:
h
– the order of the eigenvalueis_class_representative
– boolean (defaultTrue
) whether to compute instead on the conjugacy class representative
EXAMPLES:
sage: W = ReflectionGroup((1,1,3)) # optional - gap3 sage: h = W.coxeter_number() # optional - gap3 sage: for w in W: # optional - gap3 ....: 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() # optional - gap3 sage: for w in W: # optional - gap3 ....: 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]) # optional - gap3 sage: w = W.from_reduced_word([1,2,3,5]) # optional - gap3 sage: w.is_regular(4) # optional - gap3 False sage: W = ReflectionGroup(["A",3]) # optional - gap3 sage: len([w for w in W if w.is_regular(w.order())]) # optional - gap3 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 withS
equalsn
.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