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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.
The finitely presented Coxeter group W' isomorphic to the permutation
Coxeter group W, together with the isomorphism W -> W'.
The finitely presented Coxeter group W' isomorphic to the real reflection
group W
(see Chapter REFLECTION GROUPS).
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.
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.
> 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)
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'.
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.
> 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]
The finitely presented group W' isomorphic to the finitely presented
Coxeter group W, together with the isomorphism W -> W'.
The finitely presented group W' isomorphic to the permutation
Coxeter group W, together with the isomorphism W -> W'.
The finitely presented group W' isomorphic to the real reflection
group W, together with the isomorphism W -> W'
(see Chapter REFLECTION GROUPS).
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.
The permutation group W' isomorphic to the permutation
Coxeter group W, together with the isomorphism W -> W'.
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.
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