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]
<BLANKLINE>
[2]
[ 1 1]
[ 0 -1]
<BLANKLINE>
[1]
[-1 0]
[ 2 1]
<BLANKLINE>
[1, 2]
[-1 -1]
[ 2 1]
<BLANKLINE>
[2, 1]
[ 1 1]
[-2 -1]
<BLANKLINE>
[2, 1, 2]
[ 1 0]
[-2 -1]
<BLANKLINE>
[1, 2, 1]
[-1 -1]
[ 0 1]
<BLANKLINE>
[1, 2, 1, 2]
[-1 0]
[ 0 -1]
<BLANKLINE>
The standard references for actions of complex reflection groups have the matrices acting on the right, so:
sage: W.simple_reflection(1).to_matrix() # 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:
sage.structure.unique_representation.UniqueRepresentation
,sage.groups.perm_gps.permgroup.PermutationGroup_generic
A complex reflection group given as a permutation group.
See also
-
class
Element
¶ Bases:
sage.groups.perm_gps.permgroup_element.PermutationGroupElement
-
action
(vec, on_space='primal')¶ Return the image of
vec
under the action ofself
.INPUT:
vec
– vector in the basis given by the simple rooton_space
– optional (default:"primal"
) whether to act as the reflection representation on the given basis, or to act on the dual reflection representation on the dual basis
EXAMPLES:
sage: W = ReflectionGroup((3,1,2)) # optional - gap3 sage: w = W.from_reduced_word([1, 2, 1, 1, 2]) # optional - gap3 sage: for alpha in W.independent_roots(): # optional - gap3 ....: print("%s -> %s"%(alpha,w.action(alpha))) # optional - gap3 (1, 0) -> (E(3), 0) (-1, 1) -> (-E(3), E(3)^2)
-
action_on_root
(root)¶ Return the root obtained by applying
self
toroot
on the right.INPUT:
root
– the root to act on
EXAMPLES:
sage: W = ReflectionGroup(['A',2]) # optional - gap3 sage: for w in W: # optional - gap3 ....: print("%s %s"%(w.reduced_word(), # optional - gap3 ....: [w.action_on_root(beta,side="left") for beta in W.positive_roots()])) # optional - gap3 [] [(1, 0), (0, 1), (1, 1)] [2] [(1, 1), (0, -1), (1, 0)] [1] [(-1, 0), (1, 1), (0, 1)] [1, 2] [(0, 1), (-1, -1), (-1, 0)] [2, 1] [(-1, -1), (1, 0), (0, -1)] [1, 2, 1] [(0, -1), (-1, 0), (-1, -1)]
TESTS:
sage: W = ReflectionGroup(4); Phi = sorted(W.roots()) # optional - gap3 sage: all(sorted([w.action_on_root(beta) for beta in Phi]) == Phi for w in W) # optional - gap3 True
-
action_on_root_indices
(i)¶ Return the right action on the set of roots.
INPUT:
i
– index of the root to act on
EXAMPLES:
sage: W = ReflectionGroup(['A',3]) # optional - gap3 sage: w = W.w0 # optional - gap3 sage: N = len(W.roots()) # optional - gap3 sage: [w.action_on_root_indices(i) for i in range(N)] # optional - gap3 [8, 7, 6, 10, 9, 11, 2, 1, 0, 4, 3, 5] sage: W = ReflectionGroup(['A',2], reflection_index_set=['A','B','C']) # optional - gap3 sage: w = W.w0 # optional - gap3 sage: N = len(W.roots()) # optional - gap3 sage: [w.action_on_root_indices(i) for i in range(N)] # optional - gap3 [4, 3, 5, 1, 0, 2]
TESTS:
sage: W = ReflectionGroup(4) # optional - gap3 sage: N = len(W.roots()) # optional - gap3 sage: all(sorted([w.action_on_root_indices(i) for i in range(N)]) == list(range(N)) for w in W) # optional - gap3 True
-
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]
-
fix_space
()¶ Return the fix space of
self
.This is the sub vector space of the underlying vector space on which
self
acts trivially.EXAMPLES:
sage: W = ReflectionGroup((1,1,3)) # optional - gap3 sage: for w in W: # optional - gap3 ....: w.reduced_word() # optional - gap3 ....: w.fix_space() # optional - gap3 [] Vector space of degree 2 and dimension 2 over Rational Field Basis matrix: [1 0] [0 1] [2] Vector space of degree 2 and dimension 1 over Rational Field Basis matrix: [1 0] [1] Vector space of degree 2 and dimension 1 over Rational Field Basis matrix: [0 1] [1, 2] Vector space of degree 2 and dimension 0 over Rational Field Basis matrix: [] [2, 1] Vector space of degree 2 and dimension 0 over Rational Field Basis matrix: [] [1, 2, 1] Vector space of degree 2 and dimension 1 over Rational Field Basis matrix: [ 1 -1] sage: W = ReflectionGroup(23) # optional - gap3 sage: W.an_element().fix_space() # optional - gap3 Vector space of degree 3 and dimension 2 over Universal Cyclotomic Field Basis matrix: [0 1 0] [0 0 1]
-
galois_conjugates
()¶ Return all Galois conjugates of
self
.EXAMPLES:
sage: W = ReflectionGroup(4) # optional - gap3 sage: for w in W: print(w.galois_conjugates()) # optional - gap3 [[1 0] [0 1]] [[ 1 0] [ 0 E(3)], [ 1 0] [ 0 E(3)^2]] [[ 1/3*E(3) - 1/3*E(3)^2 2/3*E(3) + 1/3*E(3)^2] [ 4/3*E(3) + 2/3*E(3)^2 -1/3*E(3) - 2/3*E(3)^2], [-1/3*E(3) + 1/3*E(3)^2 1/3*E(3) + 2/3*E(3)^2] [ 2/3*E(3) + 4/3*E(3)^2 -2/3*E(3) - 1/3*E(3)^2]] [[ 1 0] [ 0 E(3)^2], [ 1 0] [ 0 E(3)]] [[ 1/3*E(3) - 1/3*E(3)^2 2/3*E(3) + 1/3*E(3)^2] [-2/3*E(3) + 2/3*E(3)^2 2/3*E(3) + 1/3*E(3)^2], [-1/3*E(3) + 1/3*E(3)^2 1/3*E(3) + 2/3*E(3)^2] [ 2/3*E(3) - 2/3*E(3)^2 1/3*E(3) + 2/3*E(3)^2]] [[ 1/3*E(3) - 1/3*E(3)^2 -1/3*E(3) + 1/3*E(3)^2] [ 4/3*E(3) + 2/3*E(3)^2 2/3*E(3) + 1/3*E(3)^2], [-1/3*E(3) + 1/3*E(3)^2 1/3*E(3) - 1/3*E(3)^2] [ 2/3*E(3) + 4/3*E(3)^2 1/3*E(3) + 2/3*E(3)^2]] [[-1/3*E(3) + 1/3*E(3)^2 1/3*E(3) + 2/3*E(3)^2] [ 2/3*E(3) + 4/3*E(3)^2 -2/3*E(3) - 1/3*E(3)^2], [ 1/3*E(3) - 1/3*E(3)^2 2/3*E(3) + 1/3*E(3)^2] [ 4/3*E(3) + 2/3*E(3)^2 -1/3*E(3) - 2/3*E(3)^2]] [[ 1/3*E(3) - 1/3*E(3)^2 2/3*E(3) + 1/3*E(3)^2] [-2/3*E(3) - 4/3*E(3)^2 -1/3*E(3) + 1/3*E(3)^2], [-1/3*E(3) + 1/3*E(3)^2 1/3*E(3) + 2/3*E(3)^2] [-4/3*E(3) - 2/3*E(3)^2 1/3*E(3) - 1/3*E(3)^2]] [[ 1/3*E(3) - 1/3*E(3)^2 -1/3*E(3) + 1/3*E(3)^2] [-2/3*E(3) + 2/3*E(3)^2 -1/3*E(3) + 1/3*E(3)^2], [-1/3*E(3) + 1/3*E(3)^2 1/3*E(3) - 1/3*E(3)^2] [ 2/3*E(3) - 2/3*E(3)^2 1/3*E(3) - 1/3*E(3)^2]] [[-1/3*E(3) + 1/3*E(3)^2 1/3*E(3) + 2/3*E(3)^2] [-4/3*E(3) - 2/3*E(3)^2 1/3*E(3) - 1/3*E(3)^2], [ 1/3*E(3) - 1/3*E(3)^2 2/3*E(3) + 1/3*E(3)^2] [-2/3*E(3) - 4/3*E(3)^2 -1/3*E(3) + 1/3*E(3)^2]] [[ 1/3*E(3) - 1/3*E(3)^2 -1/3*E(3) - 2/3*E(3)^2] [ 4/3*E(3) + 2/3*E(3)^2 -1/3*E(3) + 1/3*E(3)^2], [-1/3*E(3) + 1/3*E(3)^2 -2/3*E(3) - 1/3*E(3)^2] [ 2/3*E(3) + 4/3*E(3)^2 1/3*E(3) - 1/3*E(3)^2]] [[-1/3*E(3) + 1/3*E(3)^2 -2/3*E(3) - 1/3*E(3)^2] [ 2/3*E(3) + 4/3*E(3)^2 1/3*E(3) - 1/3*E(3)^2], [ 1/3*E(3) - 1/3*E(3)^2 -1/3*E(3) - 2/3*E(3)^2] [ 4/3*E(3) + 2/3*E(3)^2 -1/3*E(3) + 1/3*E(3)^2]] [[ 1/3*E(3) - 1/3*E(3)^2 -1/3*E(3) + 1/3*E(3)^2] [-2/3*E(3) - 4/3*E(3)^2 -1/3*E(3) - 2/3*E(3)^2], [-1/3*E(3) + 1/3*E(3)^2 1/3*E(3) - 1/3*E(3)^2] [-4/3*E(3) - 2/3*E(3)^2 -2/3*E(3) - 1/3*E(3)^2]] [[-1/3*E(3) + 1/3*E(3)^2 1/3*E(3) + 2/3*E(3)^2] [ 2/3*E(3) - 2/3*E(3)^2 1/3*E(3) + 2/3*E(3)^2], [ 1/3*E(3) - 1/3*E(3)^2 2/3*E(3) + 1/3*E(3)^2] [-2/3*E(3) + 2/3*E(3)^2 2/3*E(3) + 1/3*E(3)^2]] [[ 1/3*E(3) - 1/3*E(3)^2 -1/3*E(3) - 2/3*E(3)^2] [-2/3*E(3) + 2/3*E(3)^2 -1/3*E(3) - 2/3*E(3)^2], [-1/3*E(3) + 1/3*E(3)^2 -2/3*E(3) - 1/3*E(3)^2] [ 2/3*E(3) - 2/3*E(3)^2 -2/3*E(3) - 1/3*E(3)^2]] [[-1/3*E(3) + 1/3*E(3)^2 -2/3*E(3) - 1/3*E(3)^2] [-4/3*E(3) - 2/3*E(3)^2 1/3*E(3) + 2/3*E(3)^2], [ 1/3*E(3) - 1/3*E(3)^2 -1/3*E(3) - 2/3*E(3)^2] [-2/3*E(3) - 4/3*E(3)^2 2/3*E(3) + 1/3*E(3)^2]] [[ -1 0] [ 0 -E(3)], [ -1 0] [ 0 -E(3)^2]] [[-1/3*E(3) + 1/3*E(3)^2 1/3*E(3) - 1/3*E(3)^2] [ 2/3*E(3) + 4/3*E(3)^2 1/3*E(3) + 2/3*E(3)^2], [ 1/3*E(3) - 1/3*E(3)^2 -1/3*E(3) + 1/3*E(3)^2] [ 4/3*E(3) + 2/3*E(3)^2 2/3*E(3) + 1/3*E(3)^2]] [[ 1/3*E(3) - 1/3*E(3)^2 -1/3*E(3) - 2/3*E(3)^2] [-2/3*E(3) - 4/3*E(3)^2 2/3*E(3) + 1/3*E(3)^2], [-1/3*E(3) + 1/3*E(3)^2 -2/3*E(3) - 1/3*E(3)^2] [-4/3*E(3) - 2/3*E(3)^2 1/3*E(3) + 2/3*E(3)^2]] [[-1/3*E(3) + 1/3*E(3)^2 -2/3*E(3) - 1/3*E(3)^2] [ 2/3*E(3) - 2/3*E(3)^2 -2/3*E(3) - 1/3*E(3)^2], [ 1/3*E(3) - 1/3*E(3)^2 -1/3*E(3) - 2/3*E(3)^2] [-2/3*E(3) + 2/3*E(3)^2 -1/3*E(3) - 2/3*E(3)^2]] [[ -1 0] [ 0 -E(3)^2], [ -1 0] [ 0 -E(3)]] [[-1/3*E(3) + 1/3*E(3)^2 1/3*E(3) - 1/3*E(3)^2] [-4/3*E(3) - 2/3*E(3)^2 -2/3*E(3) - 1/3*E(3)^2], [ 1/3*E(3) - 1/3*E(3)^2 -1/3*E(3) + 1/3*E(3)^2] [-2/3*E(3) - 4/3*E(3)^2 -1/3*E(3) - 2/3*E(3)^2]] [[-1 0] [ 0 -1]] [[-1/3*E(3) + 1/3*E(3)^2 1/3*E(3) - 1/3*E(3)^2] [ 2/3*E(3) - 2/3*E(3)^2 1/3*E(3) - 1/3*E(3)^2], [ 1/3*E(3) - 1/3*E(3)^2 -1/3*E(3) + 1/3*E(3)^2] [-2/3*E(3) + 2/3*E(3)^2 -1/3*E(3) + 1/3*E(3)^2]]
-
length
()¶ Return the length of
self
in generating reflections.This is the minimal numbers of generating reflections needed to obtain
self
.EXAMPLES:
sage: W = ReflectionGroup(4) # optional - gap3 sage: for w in W: # optional - gap3 ....: print("{} {}".format(w.reduced_word(), w.length())) [] 0 [1] 1 [2] 1 [1, 1] 2 [1, 2] 2 [2, 1] 2 [2, 2] 2 [1, 1, 2] 3 [1, 2, 1] 3 [1, 2, 2] 3 [2, 1, 1] 3 [2, 2, 1] 3 [1, 1, 2, 1] 4 [1, 1, 2, 2] 4 [1, 2, 1, 1] 4 [1, 2, 2, 1] 4 [2, 1, 1, 2] 4 [2, 2, 1, 1] 4 [1, 1, 2, 1, 1] 5 [1, 1, 2, 2, 1] 5 [1, 2, 1, 1, 2] 5 [1, 2, 2, 1, 1] 5 [1, 1, 2, 1, 1, 2] 6 [1, 1, 2, 2, 1, 1] 6
-
matrix
(on_space='primal')¶ Return
self
as a matrix acting on the underlying vector space.on_space
– optional (default:"primal"
) whether to act as the reflection representation on the given basis, or to act on the dual reflection representation on the dual basis
EXAMPLES:
sage: W = ReflectionGroup((3,1,2)) # optional - gap3 sage: for w in W: # optional - gap3 ....: w.reduced_word() # optional - gap3 ....: [w.to_matrix(), w.to_matrix(on_space="dual")] # optional - gap3 [] [ [1 0] [1 0] [0 1], [0 1] ] [1] [ [E(3) 0] [E(3)^2 0] [ 0 1], [ 0 1] ] [2] [ [0 1] [0 1] [1 0], [1 0] ] [1, 1] [ [E(3)^2 0] [E(3) 0] [ 0 1], [ 0 1] ] [1, 2] [ [ 0 E(3)] [ 0 E(3)^2] [ 1 0], [ 1 0] ] [2, 1] [ [ 0 1] [ 0 1] [E(3) 0], [E(3)^2 0] ] [1, 1, 2] [ [ 0 E(3)^2] [ 0 E(3)] [ 1 0], [ 1 0] ] [1, 2, 1] [ [ 0 E(3)] [ 0 E(3)^2] [E(3) 0], [E(3)^2 0] ] [2, 1, 1] [ [ 0 1] [ 0 1] [E(3)^2 0], [E(3) 0] ] [2, 1, 2] [ [ 1 0] [ 1 0] [ 0 E(3)], [ 0 E(3)^2] ] [1, 1, 2, 1] [ [ 0 E(3)^2] [ 0 E(3)] [ E(3) 0], [E(3)^2 0] ] [1, 2, 1, 1] [ [ 0 E(3)] [ 0 E(3)^2] [E(3)^2 0], [ E(3) 0] ] [1, 2, 1, 2] [ [E(3) 0] [E(3)^2 0] [ 0 E(3)], [ 0 E(3)^2] ] [2, 1, 1, 2] [ [ 1 0] [ 1 0] [ 0 E(3)^2], [ 0 E(3)] ] [1, 1, 2, 1, 1] [ [ 0 E(3)^2] [ 0 E(3)] [E(3)^2 0], [E(3) 0] ] [1, 1, 2, 1, 2] [ [E(3)^2 0] [ E(3) 0] [ 0 E(3)], [ 0 E(3)^2] ] [1, 2, 1, 1, 2] [ [ E(3) 0] [E(3)^2 0] [ 0 E(3)^2], [ 0 E(3)] ] [1, 1, 2, 1, 1, 2] [ [E(3)^2 0] [E(3) 0] [ 0 E(3)^2], [ 0 E(3)] ]
-
reduced_word
()¶ Return a word in the simple reflections to obtain
self
.EXAMPLES:
sage: W = ReflectionGroup((5,1,1), index_set=['a'], hyperplane_index_set=['x'], reflection_index_set=['A','B','C','D']) # optional - gap3 sage: [w.reduced_word() for w in W] # optional - gap3 [[], ['a'], ['a', 'a'], ['a', 'a', 'a'], ['a', 'a', 'a', 'a']]
See also
-
reduced_word_in_reflections
()¶ Return a word in the reflections to obtain
self
.EXAMPLES:
sage: W = ReflectionGroup((5,1,1), index_set=['a'], reflection_index_set=['A','B','C','D']) # optional - gap3 sage: [w.reduced_word_in_reflections() for w in W] # optional - gap3 [[], ['A'], ['B'], ['C'], ['D']]
See also
-
reflection_eigenvalues
(is_class_representative=False)¶ Return the reflection eigenvalues of
self
.INPUT:
is_class_representative
– (default:False
) whether to first replaceself
by the representative of its conjugacy class
EXAMPLES:
sage: W = ReflectionGroup(4) # optional - gap3 sage: for w in W: w.reflection_eigenvalues() # optional - gap3 [0, 0] [1/3, 0] [1/3, 0] [2/3, 0] [1/6, 1/2] [1/6, 1/2] [2/3, 0] [1/4, 3/4] [1/4, 3/4] [1/4, 3/4] [1/4, 3/4] [1/4, 3/4] [1/3, 0] [1/2, 5/6] [1/3, 0] [1/2, 5/6] [1/2, 5/6] [1/2, 5/6] [1/6, 1/2] [2/3, 0] [1/6, 1/2] [2/3, 0] [1/2, 1/2] [1/4, 3/4]
-
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]
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to_matrix
(on_space='primal')¶ Return
self
as a matrix acting on the underlying vector space.on_space
– optional (default:"primal"
) whether to act as the reflection representation on the given basis, or to act on the dual reflection representation on the dual basis
EXAMPLES:
sage: W = ReflectionGroup((3,1,2)) # optional - gap3 sage: for w in W: # optional - gap3 ....: w.reduced_word() # optional - gap3 ....: [w.to_matrix(), w.to_matrix(on_space="dual")] # optional - gap3 [] [ [1 0] [1 0] [0 1], [0 1] ] [1] [ [E(3) 0] [E(3)^2 0] [ 0 1], [ 0 1] ] [2] [ [0 1] [0 1] [1 0], [1 0] ] [1, 1] [ [E(3)^2 0] [E(3) 0] [ 0 1], [ 0 1] ] [1, 2] [ [ 0 E(3)] [ 0 E(3)^2] [ 1 0], [ 1 0] ] [2, 1] [ [ 0 1] [ 0 1] [E(3) 0], [E(3)^2 0] ] [1, 1, 2] [ [ 0 E(3)^2] [ 0 E(3)] [ 1 0], [ 1 0] ] [1, 2, 1] [ [ 0 E(3)] [ 0 E(3)^2] [E(3) 0], [E(3)^2 0] ] [2, 1, 1] [ [ 0 1] [ 0 1] [E(3)^2 0], [E(3) 0] ] [2, 1, 2] [ [ 1 0] [ 1 0] [ 0 E(3)], [ 0 E(3)^2] ] [1, 1, 2, 1] [ [ 0 E(3)^2] [ 0 E(3)] [ E(3) 0], [E(3)^2 0] ] [1, 2, 1, 1] [ [ 0 E(3)] [ 0 E(3)^2] [E(3)^2 0], [ E(3) 0] ] [1, 2, 1, 2] [ [E(3) 0] [E(3)^2 0] [ 0 E(3)], [ 0 E(3)^2] ] [2, 1, 1, 2] [ [ 1 0] [ 1 0] [ 0 E(3)^2], [ 0 E(3)] ] [1, 1, 2, 1, 1] [ [ 0 E(3)^2] [ 0 E(3)] [E(3)^2 0], [E(3) 0] ] [1, 1, 2, 1, 2] [ [E(3)^2 0] [ E(3) 0] [ 0 E(3)], [ 0 E(3)^2] ] [1, 2, 1, 1, 2] [ [ E(3) 0] [E(3)^2 0] [ 0 E(3)^2], [ 0 E(3)] ] [1, 1, 2, 1, 1, 2] [ [E(3)^2 0] [E(3) 0] [ 0 E(3)^2], [ 0 E(3)] ]
-
to_permutation_of_roots
()¶ Return
self
as a permutation of the roots with indexing starting at \(1\).EXAMPLES:
sage: W = ReflectionGroup((1,1,3)) # optional - gap3 sage: for w in W: # optional - gap3 ....: perm = w.to_permutation_of_roots() ....: print("{} {}".format(perm, perm == w)) () True (1,3)(2,5)(4,6) True (1,4)(2,3)(5,6) True (1,6,2)(3,5,4) True (1,2,6)(3,4,5) True (1,5)(2,4)(3,6) True
-
-
ComplexReflectionGroup.
base_change_matrix
()¶ Return the base change from the standard basis of the vector space of
self
to the basis given by the independent roots ofself
.Todo
For non-well-generated groups there is a conflict with construction of the matrix for an element.
EXAMPLES:
sage: W = ReflectionGroup((1,1,3)) # optional - gap3 sage: W.base_change_matrix() # optional - gap3 [1 0] [0 1] sage: W = ReflectionGroup(23) # optional - gap3 sage: W.base_change_matrix() # optional - gap3 [1 0 0] [0 1 0] [0 0 1] sage: W = ReflectionGroup((3,1,2)) # optional - gap3 sage: W.base_change_matrix() # optional - gap3 [1 0] [1 1] sage: W = ReflectionGroup((4,2,2)) # optional - gap3 sage: W.base_change_matrix() # optional - gap3 [ 1 0] [E(4) 1]
-
ComplexReflectionGroup.
braid_relations
()¶ Return the braid relations of
self
.EXAMPLES:
sage: W = ReflectionGroup((1,1,3)) # optional - gap3 sage: W.braid_relations() # optional - gap3 [[[2, 1, 2], [1, 2, 1]]] sage: W = ReflectionGroup((2,1,3)) # optional - gap3 sage: W.braid_relations() # optional - gap3 [[[2, 1, 2, 1], [1, 2, 1, 2]], [[3, 1], [1, 3]], [[3, 2, 3], [2, 3, 2]]] sage: W = ReflectionGroup((2,2,3)) # optional - gap3 sage: W.braid_relations() # optional - gap3 [[[2, 1, 2], [1, 2, 1]], [[3, 1], [1, 3]], [[3, 2, 3], [2, 3, 2]]]
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ComplexReflectionGroup.
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]
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ComplexReflectionGroup.
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)
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ComplexReflectionGroup.
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
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ComplexReflectionGroup.
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]]
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ComplexReflectionGroup.
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)
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ComplexReflectionGroup.
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) # 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)
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ComplexReflectionGroup.
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), 'c': (1,5)(2,4)(3,6), 'b': (1,3)(2,5)(4,6)} 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)}
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ComplexReflectionGroup.
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)
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ComplexReflectionGroup.
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')
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ComplexReflectionGroup.
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)}
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ComplexReflectionGroup.
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')
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ComplexReflectionGroup.
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
TESTS:
sage: tests = [['A',3],['B',3],['F',4],(4,2,2),4,7] # optional - gap3 sage: for ty in tests: # optional - gap3 ....: W = ReflectionGroup(ty) # optional - gap3 ....: A = W.invariant_form() # optional - gap3 ....: B = W.invariant_form(brute_force=True) # optional - gap3 ....: print("{} {}".format(ty, A == B/B[0,0])) # optional - gap3 ['A', 3] True ['B', 3] True ['F', 4] True (4, 2, 2) True 4 True 7 True
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ComplexReflectionGroup.
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]
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ComplexReflectionGroup.
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
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ComplexReflectionGroup.
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
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ComplexReflectionGroup.
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
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ComplexReflectionGroup.
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)
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ComplexReflectionGroup.
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]
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ComplexReflectionGroup.
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))) # 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]
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ComplexReflectionGroup.
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): # 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): # 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]
-
ComplexReflectionGroup.
reflection_hyperplane
(i, as_linear_functional=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)
-
ComplexReflectionGroup.
reflection_hyperplanes
(as_linear_functionals=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)
-
ComplexReflectionGroup.
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')
-
ComplexReflectionGroup.
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), 'c': (1,5)(2,4)(3,6), 'b': (1,3)(2,5)(4,6)} 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)}
-
ComplexReflectionGroup.
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)]
-
ComplexReflectionGroup.
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']
-
ComplexReflectionGroup.
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
-
ComplexReflectionGroup.
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)
-
ComplexReflectionGroup.
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)}
-
ComplexReflectionGroup.
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)}
-
ComplexReflectionGroup.
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)
TESTS:
sage: W.simple_root(0) # optional - gap3 Traceback (most recent call last): ... KeyError: 0
-
ComplexReflectionGroup.
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
-
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
TESTS:
sage: from sage.categories.complex_reflection_groups import ComplexReflectionGroups sage: W = ComplexReflectionGroups().example() # optional - gap3 sage: TestSuite(W).run() # optional - gap3
-
class
Element
¶ Bases:
sage.combinat.root_system.reflection_group_complex.ComplexReflectionGroup.Element
-
is_coxeter_element
(which_primitive=1, is_class_representative=False)¶ Return
True
ifself
is a Coxeter element.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()
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())) # optional - gap3 [] 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())) # optional - gap3 [] 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 eigenvalueh
and which does not lie in any reflection hyperplane.is_class_representative
– boolean (defaultTrue
) whether to compute instead on the conjugacy class representative
EXAMPLES:
sage: W = ReflectionGroup((1,1,3)); 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): # optional - gap3 ....: w.reduced_word() # optional - gap3 [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]
-
-
IrreducibleComplexReflectionGroup.
absolute_poset
(in_unitary_group=False)¶ Return the poset induced by the absolute order of
self
as a finite lattice.INPUT:
in_unitary_group
– (default:False
) ifFalse
, the relation is given by\sigma \leq \tau
if \(l_R(\sigma) + l_R(\sigma^{-1}\tau) = l_R(\tau)\) IfTrue
, the relation is given by \(\sigma \leq \tau\) if \(\dim(\mathrm{Fix}(\sigma)) + \dim(\mathrm{Fix}(\sigma^{-1}\tau)) = \dim(\mathrm{Fix}(\tau))\).
See also
EXAMPLES:
sage: P = ReflectionGroup((1,1,3)).absolute_poset(); P # optional - gap3 Finite poset containing 6 elements sage: sorted(w.reduced_word() for w in P) # optional - gap3 [[], [1], [1, 2], [1, 2, 1], [2], [2, 1]] sage: W = ReflectionGroup(4); W # optional - gap3 Irreducible complex reflection group of rank 2 and type ST4 sage: W.absolute_poset() # optional - gap3 Finite poset containing 24 elements
-
IrreducibleComplexReflectionGroup.
elements_below_coxeter_element
(c=None)¶ Return all elements in
self
in the interval \([1,c]\) in the absolute order ofself
.This order is defined by
\[\omega \leq_R \tau \Leftrightarrow \ell_R(\omega) + \ell_R(\omega^{-1} \tau) = \ell_R(\tau),\]where \(\ell_R\) denotes the reflection length.
Note
self
is assumed to be well-generated.INPUT:
c
– (default:None
) if an elementc
is given, it is used as the maximal element in the interval; if a list is given, the union of the various maximal elements is computed
EXAMPLES:
sage: W = ReflectionGroup((1,1,3)) # optional - gap3 sage: sorted( w.reduced_word() for w in W.elements_below_coxeter_element() ) # optional - gap3 [[], [1], [1, 2], [1, 2, 1], [2]] sage: sorted( w.reduced_word() for w in W.elements_below_coxeter_element(W.from_reduced_word([2,1])) ) # optional - gap3 [[], [1], [1, 2, 1], [2], [2, 1]] sage: sorted( w.reduced_word() for w in W.elements_below_coxeter_element(W.from_reduced_word([2])) ) # optional - gap3 [[], [2]]
-
IrreducibleComplexReflectionGroup.
generalized_noncrossing_partitions
(m, c=None, positive=False)¶ Return the set of all chains of length
m
in the noncrossing partition lattice ofself
, seenoncrossing_partition_lattice()
.Note
self
is assumed to be well-generatedINPUT:
c
– (default:None
) if an elementc
inself
is given, it is used as the maximal element in the intervalpositive
– (default:False
) ifTrue
, only those generalized noncrossing partitions of full support are returned
EXAMPLES:
sage: W = ReflectionGroup((1,1,3)) # optional - gap3 sage: sorted([w.reduced_word() for w in chain] # optional - gap3 ....: for chain in W.generalized_noncrossing_partitions(2)) # optional - gap3 [[[], [], [1, 2]], [[], [1], [2]], [[], [1, 2], []], [[], [1, 2, 1], [1]], [[], [2], [1, 2, 1]], [[1], [], [2]], [[1], [2], []], [[1, 2], [], []], [[1, 2, 1], [], [1]], [[1, 2, 1], [1], []], [[2], [], [1, 2, 1]], [[2], [1, 2, 1], []]] sage: sorted([w.reduced_word() for w in chain] # optional - gap3 ....: for chain in W.generalized_noncrossing_partitions(2, positive=True)) # optional - gap3 [[[], [1, 2], []], [[], [1, 2, 1], [1]], [[1], [2], []], [[1, 2], [], []], [[1, 2, 1], [], [1]], [[1, 2, 1], [1], []], [[2], [1, 2, 1], []]]
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IrreducibleComplexReflectionGroup.
noncrossing_partition_lattice
(c=None, L=None, in_unitary_group=False)¶ Return the interval \([1,c]\) in the absolute order of
self
as a finite lattice.See also
INPUT:
c
– (default:None
) if an elementc
inself
is given, it is used as the maximal element in the intervalL
– (default:None
) if a subsetL
(must be hashable!) ofself
is given, it is used as the underlying set (only cover relations are checked)in_unitary_group
– (default:False
) ifFalse
, the relation is given by \(\sigma \leq \tau\) if \(l_R(\sigma) + l_R(\sigma^{-1}\tau) = l_R(\tau)\); ifTrue
, the relation is given by \(\sigma \leq \tau\) if \(\dim(\mathrm{Fix}(\sigma)) + \dim(\mathrm{Fix}(\sigma^{-1}\tau)) = \dim(\mathrm{Fix}(\tau))\)
EXAMPLES:
sage: W = ReflectionGroup((1,1,3)) # optional - gap3 sage: sorted( w.reduced_word() for w in W.noncrossing_partition_lattice() ) # optional - gap3 [[], [1], [1, 2], [1, 2, 1], [2]] sage: sorted( w.reduced_word() for w in W.noncrossing_partition_lattice(W.from_reduced_word([2,1])) ) # optional - gap3 [[], [1], [1, 2, 1], [2], [2, 1]] sage: sorted( w.reduced_word() for w in W.noncrossing_partition_lattice(W.from_reduced_word([2])) ) # optional - gap3 [[], [2]]
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class