# Coxeter Groups¶

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

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])
sage: W
Finite Coxeter group over Integer Ring with Coxeter matrix:
[1 3]
[3 1]

sage: W = CoxeterGroup(["A",3,1]); W
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
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") # optional - gap3
sage: 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")
sage: W
Weyl Group of type ['A', 2] (as a matrix group acting on the ambient space)

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


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

sage: W = CoxeterGroup(["H",3], implementation="reflection", base_ring=RR)
sage: W
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", index_set=['a','b'], base_ring=SR)
sage: W
Finite Coxeter group over Symbolic Ring with Coxeter matrix:
[ 1 10]
[10  1]