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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.
CoxeterGroup(GrpPermCox, N) : Cat, MonStgElt -> GrpFPCox
CoxeterGroup(N) : MonStgElt -> .
The finite or affine Coxeter group with Cartan name given by the string N
(see Section Finite and Affine Coxeter Groups).
IrreducibleCoxeterGroup(GrpPermCox, X, n) : Cat, MonStgElt, RngIntElt -> GrpPermCox
IrreducibleCoxeterGroup(X, n) : MonStgElt, RngIntElt -> .
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)
CoxeterGroup(GrpPermCox, M) : Cat, AlgMatElt -> GrpPermCox
CoxeterGroup(M) : AlgMatElt -> .
The Coxeter group with Coxeter matrix M
(see Chapter COXETER SYSTEMS).
CoxeterGroup(GrpPermCox, G) : Cat, GrphUnd -> GrpPermCox
CoxeterGroup(G) : GrphUnd -> .
The Coxeter group with Coxeter graph G
(see Chapter COXETER SYSTEMS).
CoxeterGroup(GrpPermCox, C) : Cat, AlgMatElt -> GrpPermCox
CoxeterGroup(C) : AlgMatElt -> .
The Coxeter group with Cartan matrix C
(see Chapter COXETER SYSTEMS).
CoxeterGroup(GrpPermCox, D) : Cat, GrphDir -> GrpPermCox
CoxeterGroup(D) : GrphDir -> .
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
CoxeterGroup(GrpPermCox, R) : Cat, RootSys -> GrpPermCox
CoxeterGroup(R) : RootSys -> .
CoxeterGroup(GrpFPCox, R) : Cat, RootDtm -> GrpFPCox
CoxeterGroup(GrpPermCox, R) : Cat, RootDtm -> GrpPermCox
CoxeterGroup(R) : Cat, RootDtm -> .
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)
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