It is possible to specify the category GrpFPCox or GrpPermCox when constructing a Coxeter group. If the category is not specified, then a GrpPermCox is returned for finite groups and a GrpFPCox is returned for infinite groups. If the category GrpPermCox is specified for an infinite group, an error is signalled.
The finite or affine Coxeter group with Cartan name given by the string N (see Section Finite and Affine Coxeter Groups).
The finite or affine irreducible Coxeter group with Cartan name Xn, or I2(n) if X="I" (see Section Finite and Affine Coxeter Groups).
> CoxeterGroup(GrpFPCox, "B3");
Coxeter group: Finitely presented group on 3 generators
Relations
$.1 * $.2 * $.1 = $.2 * $.1 * $.2
$.1 * $.3 = $.3 * $.1
($.2 * $.3)^2 = ($.3 * $.2)^2
$.1^2 = Id($)
$.2^2 = Id($)
$.3^2 = Id($)
> CoxeterGroup("A2B2");
Coxeter group: Permutation group acting on a set of cardinality 14
Order = 48 = 2^4 * 3
(1, 8)(2, 5)(9, 12)
(1, 5)(2, 9)(8, 12)
(3, 10)(4, 6)(11, 13)
(3, 7)(4, 11)(10, 14)
The Coxeter group with Coxeter matrix M (see Chapter COXETER SYSTEMS).
The Coxeter group with Coxeter graph G (see Chapter COXETER SYSTEMS).
The Coxeter group with Cartan matrix C (see Chapter COXETER SYSTEMS).
The Coxeter group with Dynkin digraph D (see Chapter COXETER SYSTEMS).
> M := SymmetricMatrix([ 1, 4,1, 3,4,1 ]);
> G<a,b,c> := CoxeterGroup(M);
> G;
Coxeter group: Finitely presented group on 3 generators
Relations
(a * b)^2 = (b * a)^2
a * c * a = c * a * c
(b * c)^2 = (c * b)^2
a^2 = Id($)
b^2 = Id($)
c^2 = Id($)
> M := SymmetricMatrix([ 1, 3,1, 2,3,1 ]);
> G<a,b,c> := CoxeterGroup(M);
> G;
Coxeter group: Permutation group G acting on a set of cardinality 12
Order = 24 = 2^3 * 3
(1, 7)(2, 4)(5, 6)(8, 10)(11, 12)
(1, 4)(2, 8)(3, 5)(7, 10)(9, 11)
(2, 5)(3, 9)(4, 6)(8, 11)(10, 12)
> G<a,b,c> := CoxeterGroup(GrpFPCox, M);
> G;
Coxeter group: Finitely presented group on 3 generators
Relations
a * b * a = b * a * b
a * c = c * a
b * c * b = c * b * c
a^2 = Id($)
b^2 = Id($)
c^2 = Id($)
Note that a Coxeter group does not have a unique Cartan matrix.
> C := CartanMatrix("G2");
> W := CoxeterGroup(GrpFPCox, C);
> CartanMatrix(W);
>> CartanMatrix(W);
^
Runtime error in 'CartanMatrix': Bad argument types
Argument types given: GrpFPCox
The finite Coxeter group with root system or root datum R (see Chapters ROOT SYSTEMS and ROOT DATA).
The permutation Coxeter group with roots given by the rows of the matrix A and coroots given by the rows of the matrix B. The matrices A and B must have the following properties:
- 1.
- A and B must have same number of rows and the same number of columns; they must be defined over the same field, which must be the rational field, a number field, or a cyclotomic field; the entries must be real;
- 2.
- the number of columns must be at least the number of rows; and
- 3.
- ABt must be the Cartan matrix of a finite Coxeter group.
> R := RootDatum("A3" : Isogeny := 2);
> CoxeterGroup(R);
Coxeter group: Permutation group acting on a set of cardinality 12
Order = 24 = 2^3 * 3
(1, 7)(2, 4)(5, 6)(8, 10)(11, 12)
(1, 4)(2, 8)(3, 5)(7, 10)(9, 11)
(2, 5)(3, 9)(4, 6)(8, 11)(10, 12)