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Converting Between Types of Coxeter Group

In this section, we describe functions for converting between the various descriptions of Coxeter groups available in Magma.

A Coxeter group W of type GrpMat (a reflection group), GrpPermCox (a permutation group) or GrpFPCox (a finitely presented group) can be converted to a group W' of type GrpMat, GrpPermCox, GrpPerm, GrpFPCox or GrpFP. Groups of types GrpPerm and GrpFP are respectively permutation and finitely presented groups without any attribute that identifies them as Coxeter groups. In most cases the second return value of the conversion function is an isomorphism W to W'.

Since a finitely presented Coxeter group W does not come with a built-in reflection representation, the optional parameters A, B, and C can be used to specify the representation. They are respectively the matrix whose rows are the simple roots, the matrix whose rows are the simple coroots, and the Cartan matrix. These 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.: C=AB^t must be a Cartan matrix for W.

It is not necessary to specify all three matrices, since any two of them will determine the third. If these matrices are not given, the default is to take A to be the identity and to take C to be the standard Cartan matrix described in Section Cartan Matrices.

CoxeterGroup(GrpFPCox, W) : Cat, Grp -> GrpFPCox, Map

The finitely presented Coxeter group W' isomorphic to the Coxeter group W (of type GrpPermCox or GrpMat), together with the isomorphism W to W'.

CoxeterGroup(GrpPermCox, W) : Cat, Grp -> GrpPermCox, Map

    A: Mtrx                             Default:

    B: Mtrx                             Default:

    C: Mtrx                             Default:

The permutation Coxeter group W' isomorphic to the Coxeter group W (of type GrpFPCox or GrpMat, together with the isomorphism W to W'. If W is infinite, an error is flagged.

Example `GrpCox_ConstructByGroup (H105E4)`

> W<a,b> := CoxeterGroup(GrpFPCox, "G2");
> Wp, h := CoxeterGroup(GrpPermCox, W);
> a*b;
a * b
> h(a*b);
(1, 11, 12, 7, 5, 6)(2, 4, 3, 8, 10, 9)

ReflectionGroup(W) : Grp -> GrpMat, Map

CoxeterGroup(GrpMat, W) : Cat, Grp -> GrpMat, Map

    A: Mtrx                             Default:

    B: Mtrx                             Default:

    C: Mtrx                             Default:

A reflection representation W' of the Coxeter group W (of type GrpFPCox or GrpPermCox), together with the isomorphism W to W'.
If the type of W is GrpPermCox, the optional parameters A, B, and C are ignored because in this case every permutation Coxeter group has a root system, and this determines the reflection representation.

Example `GrpCox_ReflectionGroupConversion (H105E5)`

> W<a,b,c> := CoxeterGroup(GrpFPCox, "B3");
> G, h := CoxeterGroup(GrpMat, W);
> a*b; h(a*b);
a * b
[-1 -1  0]
[ 1  0  0]
[ 0  1  1]

CoxeterGroup(GrpFP, W) : Cat, Grp -> GrpFP, Map

CoxeterGroup(GrpFP, W) : Cat, GrpPermCox -> GrpFP, Map

CoxeterGroup(GrpFP, W) : Cat, GrpMat -> GrpPermCox, Map

The finitely presented group W' isomorphic to the Coxeter group W of type GrpMat (see Chapter REFLECTION GROUPS), GrpPermCox or GrpFPCox together with the isomorphism W to W'.

CoxeterGroup(GrpPerm, W) : Cat, Grp -> GrpPerm, Map

CoxeterGroup(GrpPerm, W) : Cat, GrpMat -> GrpPermCox, Map

The permutation group W' isomorphic to the Coxeter group W of type GrpMat, GrpPermCox or GrpFPCox together with the isomorphism W to W'. If W is infinite, an error is flagged.

V2.28, 28 February 2025