Weyl Groups

AUTHORS:

• Daniel Bump (2008): initial version

• Mike Hansen (2008): initial version

• Anne Schilling (2008): initial version

• Nicolas Thiery (2008): initial version

• Volker Braun (2013): LibGAP-based matrix groups

EXAMPLES:

More examples on Weyl Groups should be added here…

The Cayley graph of the Weyl Group of type [‘A’, 3]:

sage: w = WeylGroup(['A',3])
sage: d = w.cayley_graph(); d
Digraph on 24 vertices
sage: d.show3d(color_by_label=True, edge_size=0.01, vertex_size=0.03)

The Cayley graph of the Weyl Group of type [‘D’, 4]:

sage: w = WeylGroup(['D',4])
sage: d = w.cayley_graph(); d
Digraph on 192 vertices
sage: d.show3d(color_by_label=True, edge_size=0.01, vertex_size=0.03) #long time (less than one minute)

class sage.combinat.root_system.weyl_group.ClassicalWeylSubgroup(domain, prefix)

Bases: sage.combinat.root_system.weyl_group.WeylGroup_gens

A class for Classical Weyl Subgroup of an affine Weyl Group

EXAMPLES:

sage: G = WeylGroup(["A",3,1]).classical()
sage: G
Parabolic Subgroup of the Weyl Group of type ['A', 3, 1] (as a matrix group acting on the root space)
sage: G.category()
Category of finite irreducible weyl groups
sage: G.cardinality()
24
sage: G.index_set()
(1, 2, 3)
sage: TestSuite(G).run()

Todo

implement:

• Parabolic subrootsystems

• Parabolic subgroups with a set of nodes as argument

cartan_type()

EXAMPLES:

sage: WeylGroup(['A',3,1]).classical().cartan_type()
['A', 3]
sage: WeylGroup(['A',3,1]).classical().index_set()
(1, 2, 3)

Note: won’t be needed, once the lattice will be a parabolic sub root system

simple_reflections()

EXAMPLES:

sage: WeylGroup(['A',2,1]).classical().simple_reflections()
Finite family {1: [ 1  0  0]
                  [ 1 -1  1]
                  [ 0  0  1],
               2: [ 1  0  0]
                  [ 0  1  0]
                  [ 1  1 -1]}

Note: won’t be needed, once the lattice will be a parabolic sub root system

weyl_group(prefix='hereditary')

Return the Weyl group associated to the parabolic subgroup.

EXAMPLES:

sage: WeylGroup(['A',4,1]).classical().weyl_group()
Weyl Group of type ['A', 4, 1] (as a matrix group acting on the root space)
sage: WeylGroup(['C',4,1]).classical().weyl_group()
Weyl Group of type ['C', 4, 1] (as a matrix group acting on the root space)
sage: WeylGroup(['E',8,1]).classical().weyl_group()
Weyl Group of type ['E', 8, 1] (as a matrix group acting on the root space)

sage.combinat.root_system.weyl_group.WeylGroup(x, prefix=None, implementation='matrix')

Returns the Weyl group of the root system defined by the Cartan type (or matrix) ct.

INPUT:

• x - a root system or a Cartan type (or matrix)

OPTIONAL:

• prefix – changes the representation of elements from matrices to products of simple reflections

• implementation – one of the following: * 'matrix' - as matrices acting on a root system * "permutation" - as a permutation group acting on the roots

EXAMPLES:

The following constructions yield the same result, namely a weight lattice and its corresponding Weyl group:

sage: G = WeylGroup(['F',4])
sage: L = G.domain()

or alternatively and equivalently:

sage: L = RootSystem(['F',4]).ambient_space()
sage: G = L.weyl_group()
sage: W = WeylGroup(L)

Either produces a weight lattice, with access to its roots and weights.

sage: G = WeylGroup(['F',4])
sage: G.order()
1152
sage: [s1,s2,s3,s4] = G.simple_reflections()
sage: w = s1*s2*s3*s4; w
[ 1/2  1/2  1/2  1/2]
[-1/2  1/2  1/2 -1/2]
[ 1/2  1/2 -1/2 -1/2]
[ 1/2 -1/2  1/2 -1/2]
sage: type(w) == G.element_class
True
sage: w.order()
12
sage: w.length() # length function on Weyl group
4

The default representation of Weyl group elements is as matrices. If you prefer, you may specify a prefix, in which case the elements are represented as products of simple reflections.

sage: W=WeylGroup("C3",prefix="s")
sage: [s1,s2,s3]=W.simple_reflections() # lets Sage parse its own output
sage: s2*s1*s2*s3
s1*s2*s3*s1
sage: s2*s1*s2*s3 == s1*s2*s3*s1
True
sage: (s2*s3)^2==(s3*s2)^2
True
sage: (s1*s2*s3*s1).matrix()
[ 0  0 -1]
[ 0  1  0]
[ 1  0  0]

sage: L = G.domain()
sage: fw = L.fundamental_weights(); fw
Finite family {1: (1, 1, 0, 0), 2: (2, 1, 1, 0), 3: (3/2, 1/2, 1/2, 1/2), 4: (1, 0, 0, 0)}
sage: rho = sum(fw); rho
(11/2, 5/2, 3/2, 1/2)
sage: w.action(rho) # action of G on weight lattice
(5, -1, 3, 2)

We can also do the same for arbitrary Cartan matrices:

sage: cm = CartanMatrix([[2,-5,0],[-2,2,-1],[0,-1,2]])
sage: W = WeylGroup(cm)
sage: W.gens()
(
[-1  5  0]  [ 1  0  0]  [ 1  0  0]
[ 0  1  0]  [ 2 -1  1]  [ 0  1  0]
[ 0  0  1], [ 0  0  1], [ 0  1 -1]
)
sage: s0,s1,s2 = W.gens()
sage: s1*s2*s1
[ 1  0  0]
[ 2  0 -1]
[ 2 -1  0]
sage: s2*s1*s2
[ 1  0  0]
[ 2  0 -1]
[ 2 -1  0]
sage: s0*s1*s0*s2*s0
[ 9  0 -5]
[ 2  0 -1]
[ 0  1 -1]

Same Cartan matrix, but with a prefix to display using simple reflections:

sage: W = WeylGroup(cm, prefix='s')
sage: s0,s1,s2 = W.gens()
sage: s0*s2*s1
s2*s0*s1
sage: (s1*s2)^3
1
sage: (s0*s1)^5
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1
sage: s0*s1*s2*s1*s2
s2*s0*s1
sage: s0*s1*s2*s0*s2
s0*s1*s0

class sage.combinat.root_system.weyl_group.WeylGroupElement(parent, g, check=False)

Bases: sage.groups.matrix_gps.group_element.MatrixGroupElement_gap

Class for a Weyl Group elements

action(v)

Return the action of self on the vector v.

EXAMPLES:

sage: W = WeylGroup(['A',2])
sage: s = W.simple_reflections()
sage: v = W.domain()([1,0,0])
sage: s[1].action(v)
(0, 1, 0)

sage: W = WeylGroup(RootSystem(['A',2]).root_lattice())
sage: s = W.simple_reflections()
sage: alpha = W.domain().simple_roots()
sage: s[1].action(alpha[1])
-alpha[1]

sage: W=WeylGroup(['A',2,1])
sage: alpha = W.domain().simple_roots()
sage: s = W.simple_reflections()
sage: s[1].action(alpha[1])
-alpha[1]
sage: s[1].action(alpha[0])
alpha[0] + alpha[1]

apply_simple_reflection(i, side='right')

domain()

Returns the ambient lattice associated with self.

EXAMPLES:

sage: W = WeylGroup(['A',2])
sage: s1 = W.simple_reflection(1)
sage: s1.domain()
Ambient space of the Root system of type ['A', 2]

has_descent(i, positive=False, side='right')

Test if self has a descent at position i.

An element \(w\) has a descent in position \(i\) if \(w\) is on the strict negative side of the \(i^{th}\) simple reflection hyperplane.

If positive is True, tests if it is on the strict positive side instead.

EXAMPLES:

sage: W = WeylGroup(['A',3])
sage: s = W.simple_reflections()
sage: [W.one().has_descent(i) for i in W.domain().index_set()]
[False, False, False]
sage: [s[1].has_descent(i) for i in W.domain().index_set()]
[True, False, False]
sage: [s[2].has_descent(i) for i in W.domain().index_set()]
[False, True, False]
sage: [s[3].has_descent(i) for i in W.domain().index_set()]
[False, False, True]
sage: [s[3].has_descent(i, True) for i in W.domain().index_set()]
[True, True, False]
sage: W = WeylGroup(['A',3,1])
sage: s = W.simple_reflections()
sage: [W.one().has_descent(i) for i in W.domain().index_set()]
[False, False, False, False]
sage: [s[0].has_descent(i) for i in W.domain().index_set()]
[True, False, False, False]
sage: w = s[0] * s[1]
sage: [w.has_descent(i) for i in W.domain().index_set()]
[False, True, False, False]
sage: [w.has_descent(i, side = "left") for i in W.domain().index_set()]
[True, False, False, False]
sage: w = s[0] * s[2]
sage: [w.has_descent(i) for i in W.domain().index_set()]
[True, False, True, False]
sage: [w.has_descent(i, side = "left") for i in W.domain().index_set()]
[True, False, True, False]

sage: W = WeylGroup(['A',3])
sage: W.one().has_descent(0)
True
sage: W.w0.has_descent(0)
False

has_left_descent(i)

Test if self has a left descent at position i.

EXAMPLES:

sage: W = WeylGroup(['A',3])
sage: s = W.simple_reflections()
sage: [W.one().has_left_descent(i) for i in W.domain().index_set()]
[False, False, False]
sage: [s[1].has_left_descent(i) for i in W.domain().index_set()]
[True, False, False]
sage: [s[2].has_left_descent(i) for i in W.domain().index_set()]
[False, True, False]
sage: [s[3].has_left_descent(i) for i in W.domain().index_set()]
[False, False, True]
sage: [(s[3]*s[2]).has_left_descent(i) for i in W.domain().index_set()]
[False, False, True]

has_right_descent(i)

Test if self has a right descent at position i.

EXAMPLES:

sage: W = WeylGroup(['A',3])
sage: s = W.simple_reflections()
sage: [W.one().has_right_descent(i) for i in W.domain().index_set()]
[False, False, False]
sage: [s[1].has_right_descent(i) for i in W.domain().index_set()]
[True, False, False]
sage: [s[2].has_right_descent(i) for i in W.domain().index_set()]
[False, True, False]
sage: [s[3].has_right_descent(i) for i in W.domain().index_set()]
[False, False, True]
sage: [(s[3]*s[2]).has_right_descent(i) for i in W.domain().index_set()]
[False, True, False]

to_matrix()

Return self as a matrix.

EXAMPLES:

sage: G = WeylGroup(['A',2])
sage: s1 = G.simple_reflection(1)
sage: s1.to_matrix() == s1.matrix()
True

to_permutation()

A first approximation of to_permutation …

This assumes types A,B,C,D on the ambient lattice

This further assume that the basis is indexed by 0,1,… and returns a permutation of (5,4,2,3,1) (beuargl), as a tuple

to_permutation_string()

EXAMPLES:

sage: W = WeylGroup(["A",3])
sage: s = W.simple_reflections()
sage: (s[1]*s[2]*s[3]).to_permutation_string()
'2341'

class sage.combinat.root_system.weyl_group.WeylGroup_gens(domain, prefix)

Bases: sage.structure.unique_representation.UniqueRepresentation, sage.groups.matrix_gps.finitely_generated.FinitelyGeneratedMatrixGroup_gap

EXAMPLES:

sage: G = WeylGroup(['B',3])
sage: TestSuite(G).run()
sage: cm = CartanMatrix([[2,-5,0],[-2,2,-1],[0,-1,2]])
sage: W = WeylGroup(cm)
sage: TestSuite(W).run() # long time

Element

alias of WeylGroupElement

cartan_type()

Returns the CartanType associated to self.

EXAMPLES:

sage: G = WeylGroup(['F',4])
sage: G.cartan_type()
['F', 4]

character_table()

Returns the character table as a matrix

Each row is an irreducible character. For larger tables you may preface this with a command such as gap.eval(“SizeScreen([120,40])”) in order to widen the screen.

EXAMPLES:

sage: WeylGroup(['A',3]).character_table()
CT1

     2  3  2  2  .  3
     3  1  .  .  1  .

       1a 4a 2a 3a 2b

X.1     1 -1 -1  1  1
X.2     3  1 -1  . -1
X.3     2  .  . -1  2
X.4     3 -1  1  . -1
X.5     1  1  1  1  1

classical()

If self is a Weyl group from an affine Cartan Type, this give the classical parabolic subgroup of self.

Caveat: we assume that 0 is a special node of the Dynkin diagram

TODO: extract parabolic subgroup method

EXAMPLES:

sage: G = WeylGroup(['A',3,1])
sage: G.classical()
Parabolic Subgroup of the Weyl Group of type ['A', 3, 1]
 (as a matrix group acting on the root space)
sage: WeylGroup(['A',3]).classical()
Traceback (most recent call last):
...
ValueError: classical subgroup only defined for affine types

domain()

Returns the domain of the element of self, that is the root lattice realization on which they act.

EXAMPLES:

sage: G = WeylGroup(['F',4])
sage: G.domain()
Ambient space of the Root system of type ['F', 4]
sage: G = WeylGroup(['A',3,1])
sage: G.domain()
Root space over the Rational Field of the Root system of type ['A', 3, 1]

from_morphism(f)

index_set()

Returns the index set of self.

EXAMPLES:

sage: G = WeylGroup(['F',4])
sage: G.index_set()
(1, 2, 3, 4)
sage: G = WeylGroup(['A',3,1])
sage: G.index_set()
(0, 1, 2, 3)

long_element_hardcoded()

Returns the long Weyl group element (hardcoded data)

Do we really want to keep it? There is a generic implementation which works in all cases. The hardcoded should have a better complexity (for large classical types), but there is a cache, so does this really matter?

EXAMPLES:

sage: types = [ ['A',5],['B',3],['C',3],['D',4],['G',2],['F',4],['E',6] ]
sage: [WeylGroup(t).long_element().length() for t in types]
[15, 9, 9, 12, 6, 24, 36]
sage: all( WeylGroup(t).long_element() == WeylGroup(t).long_element_hardcoded() for t in types )  # long time (17s on sage.math, 2011)
True

morphism_matrix(f)

one()

Returns the unit element of the Weyl group

EXAMPLES:

sage: W = WeylGroup(['A',3])
sage: e = W.one(); e
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
sage: type(e) == W.element_class
True

reflections()

Return the reflections of self.

The reflections of a Coxeter group \(W\) are the conjugates of the simple reflections. They are in bijection with the positive roots, for given a positive root, we may have the reflection in the hyperplane orthogonal to it. This method returns a family indexed by the positive roots taking values in the reflections. This requires self to be a finite Weyl group.

Note

Prior to trac ticket #20027, the reflections were the keys of the family and the values were the positive roots.

EXAMPLES:

sage: W = WeylGroup("B2", prefix="s")
sage: refdict = W.reflections(); refdict
Finite family {(1, -1): s1, (0, 1): s2, (1, 1): s2*s1*s2, (1, 0): s1*s2*s1}
sage: [r+refdict[r].action(r) for r in refdict.keys()]
[(0, 0), (0, 0), (0, 0), (0, 0)]

sage: W = WeylGroup(['A',2,1], prefix="s")
sage: W.reflections()
Lazy family (real root to reflection(i))_{i in
            Positive real roots of type ['A', 2, 1]}

simple_reflection(i)

Returns the \(i^{th}\) simple reflection.

EXAMPLES:

sage: G = WeylGroup(['F',4])
sage: G.simple_reflection(1)
[1 0 0 0]
[0 0 1 0]
[0 1 0 0]
[0 0 0 1]
sage: W=WeylGroup(['A',2,1])
sage: W.simple_reflection(1)
[ 1  0  0]
[ 1 -1  1]
[ 0  0  1]

simple_reflections()

Returns the simple reflections of self, as a family.

EXAMPLES:

There are the simple reflections for the symmetric group:

sage: W=WeylGroup(['A',2])
sage: s = W.simple_reflections(); s
Finite family {1: [0 1 0]
[1 0 0]
[0 0 1], 2: [1 0 0]
[0 0 1]
[0 1 0]}

As a special feature, for finite irreducible root systems, s[0] gives the reflection along the highest root:

sage: s[0]
[0 0 1]
[0 1 0]
[1 0 0]

We now look at some further examples:

sage: W=WeylGroup(['A',2,1])
sage: W.simple_reflections()
Finite family {0: [-1  1  1]
[ 0  1  0]
[ 0  0  1], 1: [ 1  0  0]
[ 1 -1  1]
[ 0  0  1], 2: [ 1  0  0]
[ 0  1  0]
[ 1  1 -1]}
sage: W = WeylGroup(['F',4])
sage: [s1,s2,s3,s4] = W.simple_reflections()
sage: w = s1*s2*s3*s4; w
[ 1/2  1/2  1/2  1/2]
[-1/2  1/2  1/2 -1/2]
[ 1/2  1/2 -1/2 -1/2]
[ 1/2 -1/2  1/2 -1/2]
sage: s4^2 == W.one()
True
sage: type(w) == W.element_class
True

unit()

Returns the unit element of the Weyl group

EXAMPLES:

sage: W = WeylGroup(['A',3])
sage: e = W.one(); e
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
sage: type(e) == W.element_class
True

class sage.combinat.root_system.weyl_group.WeylGroup_permutation(cartan_type, prefix)

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

A Weyl group given as a permutation group.

class Element

Bases: sage.combinat.root_system.reflection_group_element.RealReflectionGroupElement

cartan_type()

Return the Cartan type of self.

EXAMPLES:

sage: W = WeylGroup(['A',4], implementation="permutation")
sage: W.cartan_type()
['A', 4]

distinguished_reflections()

Return the reflections of self.

EXAMPLES:

sage: W = WeylGroup(['B',2], implementation="permutation")
sage: W.distinguished_reflections()
Finite family {1: (1,5)(2,4)(6,8), 2: (1,3)(2,6)(5,7),
               3: (2,8)(3,7)(4,6), 4: (1,7)(3,5)(4,8)}

independent_roots()

Return the simple roots of self.

EXAMPLES:

sage: W = WeylGroup(['A',4], implementation="permutation")
sage: W.simple_roots()
Finite family {1: (1, 0, 0, 0), 2: (0, 1, 0, 0),
               3: (0, 0, 1, 0), 4: (0, 0, 0, 1)}

index_set()

Return the index set of self.

EXAMPLES:

sage: W = WeylGroup(['A',4], implementation="permutation")
sage: W.index_set()
(1, 2, 3, 4)

iteration(algorithm='breadth', tracking_words=True)

Return an iterator going through all elements in self.

INPUT:

• algorithm (default: 'breadth') – must be one of the following:

  • 'breadth' - iterate over in a linear extension of the weak order

  • 'depth' - iterate by a depth-first-search

• tracking_words (default: True) – whether or not to keep track of the reduced words and store them in _reduced_word

Note

The fastest iteration is the depth first algorithm without tracking words. In particular, 'depth' is ~1.5x faster.

EXAMPLES:

sage: W = WeylGroup(["B",2], implementation="permutation")

sage: for w in W.iteration("breadth",True):
....:     print("%s %s"%(w, w._reduced_word))
() []
(1,3)(2,6)(5,7) [1]
(1,5)(2,4)(6,8) [0]
(1,7,5,3)(2,4,6,8) [0, 1]
(1,3,5,7)(2,8,6,4) [1, 0]
(2,8)(3,7)(4,6) [1, 0, 1]
(1,7)(3,5)(4,8) [0, 1, 0]
(1,5)(2,6)(3,7)(4,8) [0, 1, 0, 1]

sage: for w in W.iteration("depth", False): w
()
(1,3)(2,6)(5,7)
(1,5)(2,4)(6,8)
(1,3,5,7)(2,8,6,4)
(1,7)(3,5)(4,8)
(1,7,5,3)(2,4,6,8)
(2,8)(3,7)(4,6)
(1,5)(2,6)(3,7)(4,8)

number_of_reflections()

Return the number of reflections in self.

EXAMPLES:

sage: W = WeylGroup(['D',4], implementation="permutation")
sage: W.number_of_reflections()
12

positive_roots()

Return the positive roots of self.

EXAMPLES:

sage: W = WeylGroup(['C',3], implementation="permutation")
sage: W.positive_roots()
((1, 0, 0),
 (0, 1, 0),
 (0, 0, 1),
 (1, 1, 0),
 (0, 1, 1),
 (0, 2, 1),
 (1, 1, 1),
 (2, 2, 1),
 (1, 2, 1))

rank()

Return the rank of self.

EXAMPLES:

sage: W = WeylGroup(['A',4], implementation="permutation")
sage: W.rank()
4

reflection_index_set()

Return the index set of reflections of self.

EXAMPLES:

sage: W = WeylGroup(['A',3], implementation="permutation")
sage: W.reflection_index_set()
(1, 2, 3, 4, 5, 6)

reflections()

Return the reflections of self.

EXAMPLES:

sage: W = WeylGroup(['B',2], implementation="permutation")
sage: W.distinguished_reflections()
Finite family {1: (1,5)(2,4)(6,8), 2: (1,3)(2,6)(5,7),
               3: (2,8)(3,7)(4,6), 4: (1,7)(3,5)(4,8)}

roots()

Return the roots of self.

EXAMPLES:

sage: W = WeylGroup(['G',2], implementation="permutation")
sage: W.roots()
((1, 0),
 (0, 1),
 (1, 1),
 (3, 1),
 (2, 1),
 (3, 2),
 (-1, 0),
 (0, -1),
 (-1, -1),
 (-3, -1),
 (-2, -1),
 (-3, -2))

simple_reflection(i)

Return the i-th simple reflection of self.

EXAMPLES:

sage: W = WeylGroup(['A',4], implementation="permutation")
sage: W.simple_reflection(1)
(1,11)(2,5)(6,8)(9,10)(12,15)(16,18)(19,20)
sage: W.simple_reflections()
Finite family {1: (1,11)(2,5)(6,8)(9,10)(12,15)(16,18)(19,20),
               2: (1,5)(2,12)(3,6)(7,9)(11,15)(13,16)(17,19),
               3: (2,6)(3,13)(4,7)(5,8)(12,16)(14,17)(15,18),
               4: (3,7)(4,14)(6,9)(8,10)(13,17)(16,19)(18,20)}

simple_root_index(i)

Return the index of the simple root \(\alpha_i\).

This is the position of \(\alpha_i\) in the list of simple roots.

EXAMPLES:

sage: W = WeylGroup(['A',3], implementation="permutation")
sage: [W.simple_root_index(i) for i in W.index_set()]
[0, 1, 2]

simple_roots()

Return the simple roots of self.

EXAMPLES:

sage: W = WeylGroup(['A',4], implementation="permutation")
sage: W.simple_roots()
Finite family {1: (1, 0, 0, 0), 2: (0, 1, 0, 0),
               3: (0, 0, 1, 0), 4: (0, 0, 0, 1)}