Converting Between Types of Coxeter Group

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

Since a finitely presented Coxeter group W does not come with an in-built 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=ABt 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, GrpPermCox -> GrpFPCox, Map
The finitely presented Coxeter group W' isomorphic to the permutation Coxeter group W, together with the isomorphism W -> W'.
CoxeterGroup(GrpFPCox, W) : Cat, GrpMat -> GrpFPCox
The finitely presented Coxeter group W' isomorphic to the real reflection group W (see Chapter REFLECTION GROUPS).
CoxeterGroup(GrpPermCox, W) : Cat, GrpFPCox -> GrpPermCox, Map
    A: Mtrx                             Default: 
    B: Mtrx                             Default: 
    C: Mtrx                             Default:
The permutation Coxeter group W' isomorphic to the finitely presented Coxeter group W, together with the isomorphism W -> W'. If W is infinite, an error is flagged.
CoxeterGroup(GrpPermCox, W) : Cat, GrpMat -> GrpPermCox, Map
The permutation Coxeter group W' isomorphic to the real reflection group W, together with the isomorphism W -> W' (see Chapter REFLECTION GROUPS). 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) : GrpFPCox -> GrpMat, Map
CoxeterGroup(GrpMat, W) : Cat, GrpFPCox -> GrpPermCox, Map
    A: Mtrx                             Default: 
    B: Mtrx                             Default: 
    C: Mtrx                             Default:
A reflection group W' of the Coxeter group W, together with the isomorphism W -> W'.
ReflectionGroup(W) : GrpPermCox -> GrpMat, Map
CoxeterGroup(GrpMat, W) : Cat, GrpPermCox -> GrpMat, Map
The reflection group W' isomorphic to the permutation Coxeter group W, together with the isomorphism W -> W'. There are no optional parameters A, B, and C in this case because 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, GrpFPCox -> GrpFP, Map
The finitely presented group W' isomorphic to the finitely presented Coxeter group W, together with the isomorphism W -> W'.
CoxeterGroup(GrpFP, W) : Cat, GrpPermCox -> GrpFP, Map
The finitely presented group W' isomorphic to the permutation Coxeter group W, together with the isomorphism W -> W'.
CoxeterGroup(GrpFP, W) : Cat, GrpMat -> GrpPermCox, Map
The finitely presented group W' isomorphic to the real reflection group W, together with the isomorphism W -> W' (see Chapter REFLECTION GROUPS).
CoxeterGroup(GrpPerm, W) : Cat, GrpFPCox -> GrpPerm, Map
The permutation group W' isomorphic to the finitely presented Coxeter group W, together with the isomorphism W -> W'. If W is infinite, an error is flagged.
CoxeterGroup(GrpPerm, W) : Cat, GrpPermCox -> GrpPerm, Map
The permutation group W' isomorphic to the permutation Coxeter group W, together with the isomorphism W -> W'.
CoxeterGroup(GrpPerm, W) : Cat, GrpMat -> GrpPermCox, Map
The permutation group W' isomorphic to the real reflection group W, together with the isomorphism W -> W' (see Chapter REFLECTION GROUPS). If W is infinite, an error is flagged.
V2.28, 13 July 2023