# 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)

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 weyl groups
sage: G.cardinality()
24
sage: G.index_set()
(1, 2, 3)
sage: TestSuite(G).run()


TESTS:

sage: from sage.combinat.root_system.weyl_group import ClassicalWeylSubgroup
sage: H = ClassicalWeylSubgroup(RootSystem(["A", 3, 1]).root_space(), prefix=None)
sage: H is G
True


Caveat: the interface is likely to change. The current main application is for plots.

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)

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

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


TESTS:

sage: TestSuite(WeylGroup(["A",3])).run()
sage: TestSuite(WeylGroup(["A",3,1])).run() # long time

sage: W = WeylGroup(['A',3,1])
sage: s = W.simple_reflections()
sage: w = s[0]*s[1]*s[2]
sage: w.reduced_word()
[0, 1, 2]
sage: w = s[0]*s[2]
sage: w.reduced_word()
[2, 0]
sage: W = groups.misc.WeylGroup(['A',3,1])

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

Class for a Weyl Group elements

action(v)

Returns 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')

x.__init__(...) initializes x; see help(type(x)) for signature

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')

Tests if self has a descent at position $$i$$, that is if self 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.unit().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

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)

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

bruhat_graph(x, y)

The Bruhat graph Gamma(x,y), defined if x <= y in the Bruhat order, has as its vertices the Bruhat interval, {t | x <= t <= y}, and as its edges the pairs u, v such that u = r.v where r is a reflection, that is, a conjugate of a simple reflection.

Returns the Bruhat graph as a directed graph, with an edge u –> v if and only if u < v in the Bruhat order, and u = r.v.

See:

Carrell, The Bruhat graph of a Coxeter group, a conjecture of Deodhar, and rational smoothness of Schubert varieties. Algebraic groups and their generalizations: classical methods (University Park, PA, 1991), 53–61, Proc. Sympos. Pure Math., 56, Part 1, Amer. Math. Soc., Providence, RI, 1994.

EXAMPLES:

sage: W = WeylGroup("A3", prefix = "s")
sage: [s1,s2,s3] = W.simple_reflections()
sage: W.bruhat_graph(s1*s3,s1*s2*s3*s2*s1)
Digraph on 10 vertices

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

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]


This method used to be called lattice:

sage: G.lattice()
See http://trac.sagemath.org/8414 for details.
Root space over the Rational Field of the Root system of type ['A', 3, 1]

from_morphism(f)

x.__init__(...) initializes x; see help(type(x)) for signature

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)

lattice(*args, **kwds)

Deprecated: Use domain() instead. See trac ticket #8414 for details.

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)

x.__init__(...) initializes x; see help(type(x)) for signature

one()

Returns the unit element of the Weyl group

EXAMPLES::
sage: W = WeylGroup([‘A’,3]) sage: e = W.unit(); 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()

The reflections of 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 dictionary indexed by the reflections taking values in the positive roots. This requires self to be a finite Weyl group.

EXAMPLES:

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

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.unit()
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.unit(); e [1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1] sage: type(e) == W.element_class True

Coxeter Groups

#### Next topic

Weyl Character Rings