It is possible to specify the category GrpMat, GrpFPCox, GrpPermCox, GrpPerm or GrpFP as the first argument when constructing a Coxeter group. If the category is not specified, a permutation group of type GrpPermCox is returned for finite groups and a finitely presented group of type GrpFPCox is returned for infinite groups. If GrpPermCox is specified for an infinite group, an error is signalled.
The finite or affine Coxeter group of type grpcat with Cartan name given by the string N (see Section Finite and Affine Coxeter Groups).There are variations of this signature (see below). The second argument can be a Coxeter matrix, a Cartan matrix, a Coxeter graph, a Dynkin digraph, a root system, a root datum or, as in this case, a Cartan name given by a string.
The finite or affine irreducible Coxeter group of type grpcat 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 of type grpcat with Coxeter matrix M (see Chapter COXETER SYSTEMS).
The Coxeter group of type grpcat with Coxeter graph G (see Chapter COXETER SYSTEMS).
The Coxeter group of type grpcat with Cartan matrix C (see Chapter COXETER SYSTEMS).
The Coxeter group of type grpcat 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 of type grpcat with root system or root datum R (see Chapters ROOT SYSTEMS and ROOT DATA).
The permutation representation (or, if the group is infinite) the FP representation of the Coxeter group with simple roots given by the rows of the matrix A and simple 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 the 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 Coxeter group.
The Coxeter group of type grpcat where grpcat is GrpMat, GrpPermCox or GrpPerm). As above, the rows of A are the simple roots and the rows of B are the simple coroots.
> 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)