Coxeter Groups

sage.combinat.root_system.coxeter_group.CoxeterGroup(data, implementation='reflection', base_ring=None, index_set=None)[source]

Return an implementation of the Coxeter group given by data.

INPUT:

  • data – a Cartan type (or coercible into; see CartanType) or a Coxeter matrix or graph

  • implementation – (default: 'reflection') can be one of the following:

    • 'permutation' – as a permutation representation

    • 'matrix' – as a Weyl group (as a matrix group acting on the root space); if this is not implemented, this uses the “reflection” implementation

    • 'coxeter3' – using the coxeter3 package

    • 'reflection' – as elements in the reflection representation; see CoxeterMatrixGroup

  • base_ring – (optional) the base ring for the 'reflection' implementation

  • index_set – (optional) the index set for the 'reflection' implementation

EXAMPLES:

Now assume that data represents a Cartan type. If implementation is not specified, the reflection representation is returned:

sage: W = CoxeterGroup(["A",2]); W                                              # needs sage.libs.gap
Finite Coxeter group over Integer Ring with Coxeter matrix:
[1 3]
[3 1]

sage: W = CoxeterGroup(["A",3,1]); W                                            # needs sage.libs.gap
Coxeter group over Integer Ring with Coxeter matrix:
[1 3 2 3]
[3 1 3 2]
[2 3 1 3]
[3 2 3 1]

sage: W = CoxeterGroup(['H',3]); W                                              # needs sage.libs.gap
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(["A",Integer(2)]); W                                              # needs sage.libs.gap
Finite Coxeter group over Integer Ring with Coxeter matrix:
[1 3]
[3 1]

>>> W = CoxeterGroup(["A",Integer(3),Integer(1)]); W                                            # needs sage.libs.gap
Coxeter group over Integer Ring with Coxeter matrix:
[1 3 2 3]
[3 1 3 2]
[2 3 1 3]
[3 2 3 1]

>>> W = CoxeterGroup(['H',Integer(3)]); W                                              # needs sage.libs.gap
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]

We now use the implementation option:

sage: W = CoxeterGroup(["A",2], implementation='permutation'); W    # optional - gap3
Permutation Group with generators [(1,4)(2,3)(5,6), (1,3)(2,5)(4,6)]
sage: W.category()                                                  # optional - gap3
Join of Category of finite enumerated permutation groups
    and Category of finite Weyl groups
    and Category of well generated finite irreducible complex reflection groups

sage: W = CoxeterGroup(["A",2], implementation='matrix'); W                     # needs sage.libs.gap
Weyl Group of type ['A', 2] (as a matrix group acting on the ambient space)

sage: W = CoxeterGroup(["H",3], implementation='matrix'); 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]

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]

sage: W = CoxeterGroup(["A",4,1], implementation='permutation')                 # needs sage.libs.gap
Traceback (most recent call last):
...
ValueError: the type must be finite

sage: W = CoxeterGroup(["A",4], implementation='chevie'); W         # optional - gap3
Irreducible real reflection group of rank 4 and type A4

>>> from sage.all import *
>>> W = CoxeterGroup(["A",Integer(2)], implementation='permutation'); W    # optional - gap3
Permutation Group with generators [(1,4)(2,3)(5,6), (1,3)(2,5)(4,6)]
>>> W.category()                                                  # optional - gap3
Join of Category of finite enumerated permutation groups
    and Category of finite Weyl groups
    and Category of well generated finite irreducible complex reflection groups

>>> W = CoxeterGroup(["A",Integer(2)], implementation='matrix'); W                     # needs sage.libs.gap
Weyl Group of type ['A', 2] (as a matrix group acting on the ambient space)

>>> W = CoxeterGroup(["H",Integer(3)], implementation='matrix'); 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]

>>> 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]

>>> W = CoxeterGroup(["A",Integer(4),Integer(1)], implementation='permutation')                 # needs sage.libs.gap
Traceback (most recent call last):
...
ValueError: the type must be finite

>>> W = CoxeterGroup(["A",Integer(4)], implementation='chevie'); W         # optional - gap3
Irreducible real reflection group of rank 4 and type A4

We use the different options for the “reflection” implementation:

sage: W = CoxeterGroup(["H",3], implementation='reflection', base_ring=RR); W   # needs sage.libs.gap
Finite Coxeter group over Real Field with 53 bits of precision with Coxeter matrix:
[1 3 2]
[3 1 5]
[2 5 1]
sage: W = CoxeterGroup([[1,10],[10,1]], implementation='reflection',            # needs sage.symbolics
....:                  index_set=['a','b'], base_ring=SR); W
Finite Coxeter group over Symbolic Ring with Coxeter matrix:
[ 1 10]
[10  1]

>>> from sage.all import *
>>> W = CoxeterGroup(["H",Integer(3)], implementation='reflection', base_ring=RR); W   # needs sage.libs.gap
Finite Coxeter group over Real Field with 53 bits of precision with Coxeter matrix:
[1 3 2]
[3 1 5]
[2 5 1]
>>> W = CoxeterGroup([[Integer(1),Integer(10)],[Integer(10),Integer(1)]], implementation='reflection',            # needs sage.symbolics
...                  index_set=['a','b'], base_ring=SR); W
Finite Coxeter group over Symbolic Ring with Coxeter matrix:
[ 1 10]
[10  1]