Coxeter Groups As Matrix Groups#

This implements a general Coxeter group as a matrix group by using the reflection representation.

AUTHORS:

  • Travis Scrimshaw (2013-08-28): Initial version

class sage.groups.matrix_gps.coxeter_group.CoxeterMatrixGroup(coxeter_matrix, base_ring, index_set)[source]#

Bases: UniqueRepresentation, FinitelyGeneratedMatrixGroup_generic

A Coxeter group represented as a matrix group.

Let \((W, S)\) be a Coxeter system. We construct a vector space \(V\) over \(\RR\) with a basis of \(\{ \alpha_s \}_{s \in S}\) and inner product

\[B(\alpha_s, \alpha_t) = -\cos\left( \frac{\pi}{m_{st}} \right)\]

where we have \(B(\alpha_s, \alpha_t) = -1\) if \(m_{st} = \infty\). Next we define a representation \(\sigma_s : V \to V\) by

\[\sigma_s \lambda = \lambda - 2 B(\alpha_s, \lambda) \alpha_s.\]

This representation is faithful so we can represent the Coxeter group \(W\) by the set of matrices \(\sigma_s\) acting on \(V\).

INPUT:

  • data – a Coxeter matrix or graph or a Cartan type

  • base_ring – (default: the universal cyclotomic field or a number field) the base ring which contains all values \(\cos(\pi/m_{ij})\) where \((m_{ij})_{ij}\) is the Coxeter matrix

  • index_set – (optional) an indexing set for the generators

For finite Coxeter groups, the default base ring is taken to be \(\QQ\) or a quadratic number field when possible.

For more on creating Coxeter groups, see CoxeterGroup().

Todo

Currently the label \(\infty\) is implemented as \(-1\) in the Coxeter matrix.

EXAMPLES:

We can create Coxeter groups from Coxeter matrices:

sage: # needs sage.libs.gap sage.rings.number_field
sage: W = CoxeterGroup([[1, 6, 3], [6, 1, 10], [3, 10, 1]]); W
Coxeter group over Universal Cyclotomic Field with Coxeter matrix:
[ 1  6  3]
[ 6  1 10]
[ 3 10  1]
sage: W.gens()
(
[                 -1 -E(12)^7 + E(12)^11                   1]
[                  0                   1                   0]
[                  0                   0                   1],

[                  1                   0                   0]
[-E(12)^7 + E(12)^11                  -1     E(20) - E(20)^9]
[                  0                   0                   1],

[              1               0               0]
[              0               1               0]
[              1 E(20) - E(20)^9              -1]
)

sage: m = matrix([[1,3,3,3], [3,1,3,2], [3,3,1,2], [3,2,2,1]])
sage: W = CoxeterGroup(m)
sage: W.gens()
(
[-1  1  1  1]  [ 1  0  0  0]  [ 1  0  0  0]  [ 1  0  0  0]
[ 0  1  0  0]  [ 1 -1  1  0]  [ 0  1  0  0]  [ 0  1  0  0]
[ 0  0  1  0]  [ 0  0  1  0]  [ 1  1 -1  0]  [ 0  0  1  0]
[ 0  0  0  1], [ 0  0  0  1], [ 0  0  0  1], [ 1  0  0 -1]
)
sage: a,b,c,d = W.gens()
sage: (a*b*c)^3
[ 5  1 -5  7]
[ 5  0 -4  5]
[ 4  1 -4  4]
[ 0  0  0  1]
sage: (a*b)^3
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
sage: b*d == d*b
True
sage: a*c*a == c*a*c
True

>>> from sage.all import *
>>> # needs sage.libs.gap sage.rings.number_field
>>> W = CoxeterGroup([[Integer(1), Integer(6), Integer(3)], [Integer(6), Integer(1), Integer(10)], [Integer(3), Integer(10), Integer(1)]]); W
Coxeter group over Universal Cyclotomic Field with Coxeter matrix:
[ 1  6  3]
[ 6  1 10]
[ 3 10  1]
>>> W.gens()
(
[                 -1 -E(12)^7 + E(12)^11                   1]
[                  0                   1                   0]
[                  0                   0                   1],
<BLANKLINE>
[                  1                   0                   0]
[-E(12)^7 + E(12)^11                  -1     E(20) - E(20)^9]
[                  0                   0                   1],
<BLANKLINE>
[              1               0               0]
[              0               1               0]
[              1 E(20) - E(20)^9              -1]
)

>>> m = matrix([[Integer(1),Integer(3),Integer(3),Integer(3)], [Integer(3),Integer(1),Integer(3),Integer(2)], [Integer(3),Integer(3),Integer(1),Integer(2)], [Integer(3),Integer(2),Integer(2),Integer(1)]])
>>> W = CoxeterGroup(m)
>>> W.gens()
(
[-1  1  1  1]  [ 1  0  0  0]  [ 1  0  0  0]  [ 1  0  0  0]
[ 0  1  0  0]  [ 1 -1  1  0]  [ 0  1  0  0]  [ 0  1  0  0]
[ 0  0  1  0]  [ 0  0  1  0]  [ 1  1 -1  0]  [ 0  0  1  0]
[ 0  0  0  1], [ 0  0  0  1], [ 0  0  0  1], [ 1  0  0 -1]
)
>>> a,b,c,d = W.gens()
>>> (a*b*c)**Integer(3)
[ 5  1 -5  7]
[ 5  0 -4  5]
[ 4  1 -4  4]
[ 0  0  0  1]
>>> (a*b)**Integer(3)
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
>>> b*d == d*b
True
>>> a*c*a == c*a*c
True

We can create the matrix representation over different base rings and with different index sets. Note that the base ring must contain all \(2*\cos(\pi/m_{ij})\) where \((m_{ij})_{ij}\) is the Coxeter matrix:

sage: W = CoxeterGroup(m, base_ring=RR, index_set=['a','b','c','d'])
sage: W.base_ring()
Real Field with 53 bits of precision
sage: W.index_set()
('a', 'b', 'c', 'd')

sage: CoxeterGroup(m, base_ring=ZZ)
Coxeter group over Integer Ring with Coxeter matrix:
[1 3 3 3]
[3 1 3 2]
[3 3 1 2]
[3 2 2 1]
sage: CoxeterGroup([[1,4],[4,1]], base_ring=QQ)                                 # needs sage.symbolic
Traceback (most recent call last):
...
TypeError: unable to convert sqrt(2) to a rational

>>> from sage.all import *
>>> W = CoxeterGroup(m, base_ring=RR, index_set=['a','b','c','d'])
>>> W.base_ring()
Real Field with 53 bits of precision
>>> W.index_set()
('a', 'b', 'c', 'd')

>>> CoxeterGroup(m, base_ring=ZZ)
Coxeter group over Integer Ring with Coxeter matrix:
[1 3 3 3]
[3 1 3 2]
[3 3 1 2]
[3 2 2 1]
>>> CoxeterGroup([[Integer(1),Integer(4)],[Integer(4),Integer(1)]], base_ring=QQ)                                 # needs sage.symbolic
Traceback (most recent call last):
...
TypeError: unable to convert sqrt(2) to a rational

Using the well-known conversion between Coxeter matrices and Coxeter graphs, we can input a Coxeter graph. Following the standard convention, edges with no label (i.e. labelled by None) are treated as 3:

sage: # needs sage.libs.gap sage.rings.number_field
sage: G = Graph([(0,3,None), (1,3,15), (2,3,7), (0,1,3)])
sage: W = CoxeterGroup(G); W
Coxeter group over Universal Cyclotomic Field with Coxeter matrix:
[ 1  3  2  3]
[ 3  1  2 15]
[ 2  2  1  7]
[ 3 15  7  1]
sage: G2 = W.coxeter_diagram()
sage: CoxeterGroup(G2) is W
True

>>> from sage.all import *
>>> # needs sage.libs.gap sage.rings.number_field
>>> G = Graph([(Integer(0),Integer(3),None), (Integer(1),Integer(3),Integer(15)), (Integer(2),Integer(3),Integer(7)), (Integer(0),Integer(1),Integer(3))])
>>> W = CoxeterGroup(G); W
Coxeter group over Universal Cyclotomic Field with Coxeter matrix:
[ 1  3  2  3]
[ 3  1  2 15]
[ 2  2  1  7]
[ 3 15  7  1]
>>> G2 = W.coxeter_diagram()
>>> CoxeterGroup(G2) is W
True

Because there currently is no class for \(\ZZ \cup \{ \infty \}\), labels of \(\infty\) are given by \(-1\) in the Coxeter matrix:

sage: # needs sage.libs.gap sage.rings.number_field
sage: G = Graph([(0,1,None), (1,2,4), (0,2,oo)])
sage: W = CoxeterGroup(G)
sage: W.coxeter_matrix()
[ 1  3 -1]
[ 3  1  4]
[-1  4  1]

>>> from sage.all import *
>>> # needs sage.libs.gap sage.rings.number_field
>>> G = Graph([(Integer(0),Integer(1),None), (Integer(1),Integer(2),Integer(4)), (Integer(0),Integer(2),oo)])
>>> W = CoxeterGroup(G)
>>> W.coxeter_matrix()
[ 1  3 -1]
[ 3  1  4]
[-1  4  1]

We can also create Coxeter groups from Cartan types using the implementation keyword:

sage: W = CoxeterGroup(['D',5], implementation="reflection"); W
Finite Coxeter group over Integer Ring with Coxeter matrix:
[1 3 2 2 2]
[3 1 3 2 2]
[2 3 1 3 3]
[2 2 3 1 2]
[2 2 3 2 1]
sage: W = CoxeterGroup(['H',3], implementation="reflection"); W                 # needs sage.libs.gap sage.rings.number_field
Finite Coxeter group over
 Number Field in a with defining polynomial x^2 - 5 with a = 2.236067977499790?
 with Coxeter matrix:
 [1 3 2]
 [3 1 5]
 [2 5 1]

>>> from sage.all import *
>>> W = CoxeterGroup(['D',Integer(5)], implementation="reflection"); W
Finite Coxeter group over Integer Ring with Coxeter matrix:
[1 3 2 2 2]
[3 1 3 2 2]
[2 3 1 3 3]
[2 2 3 1 2]
[2 2 3 2 1]
>>> W = CoxeterGroup(['H',Integer(3)], implementation="reflection"); W                 # needs sage.libs.gap sage.rings.number_field
Finite Coxeter group over
 Number Field in a with defining polynomial x^2 - 5 with a = 2.236067977499790?
 with Coxeter matrix:
 [1 3 2]
 [3 1 5]
 [2 5 1]
class Element[source]#

Bases: MatrixGroupElement_generic

A Coxeter group element.

action_on_root_indices(i, side='left')[source]#

Return the action on the set of roots.

The roots are ordered as in the output of the method roots().

EXAMPLES:

sage: W = CoxeterGroup(['A',3], implementation="reflection")
sage: w = W.w0
sage: w.action_on_root_indices(0)
11

>>> from sage.all import *
>>> W = CoxeterGroup(['A',Integer(3)], implementation="reflection")
>>> w = W.w0
>>> w.action_on_root_indices(Integer(0))
11
canonical_matrix()[source]#

Return the matrix of self in the canonical faithful representation, which is self as a matrix.

EXAMPLES:

sage: W = CoxeterGroup(['A',3], implementation="reflection")
sage: a,b,c = W.gens()
sage: elt = a*b*c
sage: elt.canonical_matrix()
[ 0  0 -1]
[ 1  0 -1]
[ 0  1 -1]

>>> from sage.all import *
>>> W = CoxeterGroup(['A',Integer(3)], implementation="reflection")
>>> a,b,c = W.gens()
>>> elt = a*b*c
>>> elt.canonical_matrix()
[ 0  0 -1]
[ 1  0 -1]
[ 0  1 -1]
descents(side='right', index_set=None, positive=False)[source]#

Return the descents of self, as a list of elements of the index_set.

INPUT:

  • index_set – (default: all of them) a subset (as a list or iterable) of the nodes of the Dynkin diagram

  • side – (default: 'right') 'left' or 'right'

  • positive – (default: False) boolean

EXAMPLES:

sage: W = CoxeterGroup(['A',3], implementation="reflection")
sage: a,b,c = W.gens()
sage: elt = b*a*c
sage: elt.descents()
[1, 3]
sage: elt.descents(positive=True)
[2]
sage: elt.descents(index_set=[1,2])
[1]
sage: elt.descents(side='left')
[2]

>>> from sage.all import *
>>> W = CoxeterGroup(['A',Integer(3)], implementation="reflection")
>>> a,b,c = W.gens()
>>> elt = b*a*c
>>> elt.descents()
[1, 3]
>>> elt.descents(positive=True)
[2]
>>> elt.descents(index_set=[Integer(1),Integer(2)])
[1]
>>> elt.descents(side='left')
[2]
first_descent(side='right', index_set=None, positive=False)[source]#

Return the first left (resp. right) descent of self, as ane element of index_set, or None if there is none.

See descents() for a description of the options.

EXAMPLES:

sage: W = CoxeterGroup(['A',3], implementation="reflection")
sage: a,b,c = W.gens()
sage: elt = b*a*c
sage: elt.first_descent()
1
sage: elt.first_descent(side='left')
2

>>> from sage.all import *
>>> W = CoxeterGroup(['A',Integer(3)], implementation="reflection")
>>> a,b,c = W.gens()
>>> elt = b*a*c
>>> elt.first_descent()
1
>>> elt.first_descent(side='left')
2
has_right_descent(i)[source]#

Return whether i is a right descent of self.

A Coxeter system \((W, S)\) has a root system defined as \(\{ w(\alpha_s) \}_{w \in W}\) and we define the positive (resp. negative) roots \(\alpha = \sum_{s \in S} c_s \alpha_s\) by all \(c_s \geq 0\) (resp. \(c_s \leq 0\)). In particular, we note that if \(\ell(w s) > \ell(w)\) then \(w(\alpha_s) > 0\) and if \(\ell(ws) < \ell(w)\) then \(w(\alpha_s) < 0\). Thus \(i \in I\) is a right descent if \(w(\alpha_{s_i}) < 0\) or equivalently if the matrix representing \(w\) has all entries of the \(i\)-th column being non-positive.

INPUT:

  • i – an element in the index set

EXAMPLES:

sage: W = CoxeterGroup(['A',3], implementation="reflection")
sage: a,b,c = W.gens()
sage: elt = b*a*c
sage: [elt.has_right_descent(i) for i in [1, 2, 3]]
[True, False, True]

>>> from sage.all import *
>>> W = CoxeterGroup(['A',Integer(3)], implementation="reflection")
>>> a,b,c = W.gens()
>>> elt = b*a*c
>>> [elt.has_right_descent(i) for i in [Integer(1), Integer(2), Integer(3)]]
[True, False, True]
bilinear_form()[source]#

Return the bilinear form associated to self.

Given a Coxeter group \(G\) with Coxeter matrix \(M = (m_{ij})_{ij}\), the associated bilinear form \(A = (a_{ij})_{ij}\) is given by

\[a_{ij} = -\cos\left( \frac{\pi}{m_{ij}} \right).\]

If \(A\) is positive definite, then \(G\) is of finite type (and so the associated Coxeter group is a finite group). If \(A\) is positive semidefinite, then \(G\) is affine type.

EXAMPLES:

sage: W = CoxeterGroup(['D',4])
sage: W.bilinear_form()                                                     # needs sage.symbolic
[   1 -1/2    0    0]
[-1/2    1 -1/2 -1/2]
[   0 -1/2    1    0]
[   0 -1/2    0    1]

>>> from sage.all import *
>>> W = CoxeterGroup(['D',Integer(4)])
>>> W.bilinear_form()                                                     # needs sage.symbolic
[   1 -1/2    0    0]
[-1/2    1 -1/2 -1/2]
[   0 -1/2    1    0]
[   0 -1/2    0    1]
canonical_representation()[source]#

Return the canonical faithful representation of self, which is self.

EXAMPLES:

sage: W = CoxeterGroup([[1,3],[3,1]])
sage: W.canonical_representation() is W
True

>>> from sage.all import *
>>> W = CoxeterGroup([[Integer(1),Integer(3)],[Integer(3),Integer(1)]])
>>> W.canonical_representation() is W
True
coxeter_matrix()[source]#

Return the Coxeter matrix of self.

EXAMPLES:

sage: W = CoxeterGroup([[1,3],[3,1]])
sage: W.coxeter_matrix()
[1 3]
[3 1]
sage: W = CoxeterGroup(['H',3])                                             # needs sage.libs.gap sage.rings.number_field
sage: W.coxeter_matrix()                                                    # needs sage.libs.gap sage.rings.number_field
[1 3 2]
[3 1 5]
[2 5 1]

>>> from sage.all import *
>>> W = CoxeterGroup([[Integer(1),Integer(3)],[Integer(3),Integer(1)]])
>>> W.coxeter_matrix()
[1 3]
[3 1]
>>> W = CoxeterGroup(['H',Integer(3)])                                             # needs sage.libs.gap sage.rings.number_field
>>> W.coxeter_matrix()                                                    # needs sage.libs.gap sage.rings.number_field
[1 3 2]
[3 1 5]
[2 5 1]
fundamental_weight(i)[source]#

Return the fundamental weight with index i.

See also

fundamental_weights()

EXAMPLES:

sage: W = CoxeterGroup(['A',3], implementation='reflection')
sage: W.fundamental_weight(1)                                               # needs sage.symbolic
(3/2, 1, 1/2)

>>> from sage.all import *
>>> W = CoxeterGroup(['A',Integer(3)], implementation='reflection')
>>> W.fundamental_weight(Integer(1))                                               # needs sage.symbolic
(3/2, 1, 1/2)
fundamental_weights()[source]#

Return the fundamental weights for self.

This is the dual basis to the basis of simple roots.

The base ring must be a field.

See also

fundamental_weight()

EXAMPLES:

sage: W = CoxeterGroup(['A',3], implementation='reflection')
sage: W.fundamental_weights()                                               # needs sage.symbolic
Finite family {1: (3/2, 1, 1/2), 2: (1, 2, 1), 3: (1/2, 1, 3/2)}

>>> from sage.all import *
>>> W = CoxeterGroup(['A',Integer(3)], implementation='reflection')
>>> W.fundamental_weights()                                               # needs sage.symbolic
Finite family {1: (3/2, 1, 1/2), 2: (1, 2, 1), 3: (1/2, 1, 3/2)}
is_commutative()[source]#

Return whether self is commutative.

EXAMPLES:

sage: CoxeterGroup(['A', 2]).is_commutative()
False
sage: W = CoxeterGroup(['I',2])
sage: W.is_commutative()
True

>>> from sage.all import *
>>> CoxeterGroup(['A', Integer(2)]).is_commutative()
False
>>> W = CoxeterGroup(['I',Integer(2)])
>>> W.is_commutative()
True
is_finite()[source]#

Return True if this group is finite.

EXAMPLES:

sage: # needs sage.libs.gap sage.rings.number_field
sage: [l for l in range(2, 9) if
....:  CoxeterGroup([[1,3,2],[3,1,l],[2,l,1]]).is_finite()]
[2, 3, 4, 5]
sage: [l for l in range(2, 9) if
....:  CoxeterGroup([[1,3,2,2],[3,1,l,2],[2,l,1,3],[2,2,3,1]]).is_finite()]
[2, 3, 4]
sage: [l for l in range(2, 9) if
....:  CoxeterGroup([[1,3,2,2,2], [3,1,3,3,2], [2,3,1,2,2],
....:                [2,3,2,1,l], [2,2,2,l,1]]).is_finite()]
[2, 3]
sage: [l for l in range(2, 9) if
....:  CoxeterGroup([[1,3,2,2,2], [3,1,2,3,3], [2,2,1,l,2],
....:                [2,3,l,1,2], [2,3,2,2,1]]).is_finite()]
[2, 3]
sage: [l for l in range(2, 9) if
....:  CoxeterGroup([[1,3,2,2,2,2], [3,1,l,2,2,2], [2,l,1,3,l,2],
....:                [2,2,3,1,2,2], [2,2,l,2,1,3], [2,2,2,2,3,1]]).is_finite()]
[2, 3]

>>> from sage.all import *
>>> # needs sage.libs.gap sage.rings.number_field
>>> [l for l in range(Integer(2), Integer(9)) if
...  CoxeterGroup([[Integer(1),Integer(3),Integer(2)],[Integer(3),Integer(1),l],[Integer(2),l,Integer(1)]]).is_finite()]
[2, 3, 4, 5]
>>> [l for l in range(Integer(2), Integer(9)) if
...  CoxeterGroup([[Integer(1),Integer(3),Integer(2),Integer(2)],[Integer(3),Integer(1),l,Integer(2)],[Integer(2),l,Integer(1),Integer(3)],[Integer(2),Integer(2),Integer(3),Integer(1)]]).is_finite()]
[2, 3, 4]
>>> [l for l in range(Integer(2), Integer(9)) if
...  CoxeterGroup([[Integer(1),Integer(3),Integer(2),Integer(2),Integer(2)], [Integer(3),Integer(1),Integer(3),Integer(3),Integer(2)], [Integer(2),Integer(3),Integer(1),Integer(2),Integer(2)],
...                [Integer(2),Integer(3),Integer(2),Integer(1),l], [Integer(2),Integer(2),Integer(2),l,Integer(1)]]).is_finite()]
[2, 3]
>>> [l for l in range(Integer(2), Integer(9)) if
...  CoxeterGroup([[Integer(1),Integer(3),Integer(2),Integer(2),Integer(2)], [Integer(3),Integer(1),Integer(2),Integer(3),Integer(3)], [Integer(2),Integer(2),Integer(1),l,Integer(2)],
...                [Integer(2),Integer(3),l,Integer(1),Integer(2)], [Integer(2),Integer(3),Integer(2),Integer(2),Integer(1)]]).is_finite()]
[2, 3]
>>> [l for l in range(Integer(2), Integer(9)) if
...  CoxeterGroup([[Integer(1),Integer(3),Integer(2),Integer(2),Integer(2),Integer(2)], [Integer(3),Integer(1),l,Integer(2),Integer(2),Integer(2)], [Integer(2),l,Integer(1),Integer(3),l,Integer(2)],
...                [Integer(2),Integer(2),Integer(3),Integer(1),Integer(2),Integer(2)], [Integer(2),Integer(2),l,Integer(2),Integer(1),Integer(3)], [Integer(2),Integer(2),Integer(2),Integer(2),Integer(3),Integer(1)]]).is_finite()]
[2, 3]
order()[source]#

Return the order of self.

If the Coxeter group is finite, this uses an iterator.

EXAMPLES:

sage: # needs sage.rings.number_field
sage: W = CoxeterGroup([[1,3],[3,1]])
sage: W.order()
6
sage: W = CoxeterGroup([[1,-1],[-1,1]])                                     # needs sage.libs.gap
sage: W.order()                                                             # needs sage.libs.gap
+Infinity

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> W = CoxeterGroup([[Integer(1),Integer(3)],[Integer(3),Integer(1)]])
>>> W.order()
6
>>> W = CoxeterGroup([[Integer(1),-Integer(1)],[-Integer(1),Integer(1)]])                                     # needs sage.libs.gap
>>> W.order()                                                             # needs sage.libs.gap
+Infinity
positive_roots()[source]#

Return the positive roots.

These are roots in the Coxeter sense, that all have the same norm. They are given by their coefficients in the base of simple roots, also taken to have all the same norm.

See also

reflections()

EXAMPLES:

sage: W = CoxeterGroup(['A',3], implementation='reflection')
sage: W.positive_roots()
((1, 0, 0), (1, 1, 0), (0, 1, 0), (1, 1, 1), (0, 1, 1), (0, 0, 1))

sage: # needs sage.libs.gap sage.rings.number_field
sage: W = CoxeterGroup(['I',5], implementation='reflection')
sage: W.positive_roots()
((1, 0),
 (-E(5)^2 - E(5)^3, 1),
 (-E(5)^2 - E(5)^3, -E(5)^2 - E(5)^3),
 (1, -E(5)^2 - E(5)^3),
 (0, 1))

>>> from sage.all import *
>>> W = CoxeterGroup(['A',Integer(3)], implementation='reflection')
>>> W.positive_roots()
((1, 0, 0), (1, 1, 0), (0, 1, 0), (1, 1, 1), (0, 1, 1), (0, 0, 1))

>>> # needs sage.libs.gap sage.rings.number_field
>>> W = CoxeterGroup(['I',Integer(5)], implementation='reflection')
>>> W.positive_roots()
((1, 0),
 (-E(5)^2 - E(5)^3, 1),
 (-E(5)^2 - E(5)^3, -E(5)^2 - E(5)^3),
 (1, -E(5)^2 - E(5)^3),
 (0, 1))
reflections()[source]#

Return the set of reflections.

The order is the one given by positive_roots().

EXAMPLES:

sage: W = CoxeterGroup(['A',2], implementation='reflection')
sage: list(W.reflections())
[
[-1  1]  [ 0 -1]  [ 1  0]
[ 0  1], [-1  0], [ 1 -1]
]

>>> from sage.all import *
>>> W = CoxeterGroup(['A',Integer(2)], implementation='reflection')
>>> list(W.reflections())
[
[-1  1]  [ 0 -1]  [ 1  0]
[ 0  1], [-1  0], [ 1 -1]
]
roots()[source]#

Return the roots.

These are roots in the Coxeter sense, that all have the same norm. They are given by their coefficients in the base of simple roots, also taken to have all the same norm.

The positive roots are listed first, then the negative roots in the same order. The order is the one given by roots().

EXAMPLES:

sage: W = CoxeterGroup(['A',3], implementation='reflection')
sage: W.roots()
((1, 0, 0),
 (1, 1, 0),
 (0, 1, 0),
 (1, 1, 1),
 (0, 1, 1),
 (0, 0, 1),
 (-1, 0, 0),
 (-1, -1, 0),
 (0, -1, 0),
 (-1, -1, -1),
 (0, -1, -1),
 (0, 0, -1))

sage: # needs sage.libs.gap sage.rings.number_field
sage: W = CoxeterGroup(['I',5], implementation='reflection')
sage: len(W.roots())
10

>>> from sage.all import *
>>> W = CoxeterGroup(['A',Integer(3)], implementation='reflection')
>>> W.roots()
((1, 0, 0),
 (1, 1, 0),
 (0, 1, 0),
 (1, 1, 1),
 (0, 1, 1),
 (0, 0, 1),
 (-1, 0, 0),
 (-1, -1, 0),
 (0, -1, 0),
 (-1, -1, -1),
 (0, -1, -1),
 (0, 0, -1))

>>> # needs sage.libs.gap sage.rings.number_field
>>> W = CoxeterGroup(['I',Integer(5)], implementation='reflection')
>>> len(W.roots())
10
simple_reflection(i)[source]#

Return the simple reflection \(s_i\).

INPUT:

  • i – an element from the index set

EXAMPLES:

sage: W = CoxeterGroup(['A',3], implementation="reflection")
sage: W.simple_reflection(1)
[-1  1  0]
[ 0  1  0]
[ 0  0  1]
sage: W.simple_reflection(2)
[ 1  0  0]
[ 1 -1  1]
[ 0  0  1]
sage: W.simple_reflection(3)
[ 1  0  0]
[ 0  1  0]
[ 0  1 -1]

>>> from sage.all import *
>>> W = CoxeterGroup(['A',Integer(3)], implementation="reflection")
>>> W.simple_reflection(Integer(1))
[-1  1  0]
[ 0  1  0]
[ 0  0  1]
>>> W.simple_reflection(Integer(2))
[ 1  0  0]
[ 1 -1  1]
[ 0  0  1]
>>> W.simple_reflection(Integer(3))
[ 1  0  0]
[ 0  1  0]
[ 0  1 -1]
simple_root_index(i)[source]#

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

This is the position of \(\alpha_i\) in the list of all roots as given be roots().

EXAMPLES:

sage: W = CoxeterGroup(['A',3], implementation='reflection')
sage: [W.simple_root_index(i) for i in W.index_set()]
[0, 2, 5]

>>> from sage.all import *
>>> W = CoxeterGroup(['A',Integer(3)], implementation='reflection')
>>> [W.simple_root_index(i) for i in W.index_set()]
[0, 2, 5]