Constructing Coxeter Groups

For many of the intrinsics that construct Coxeter groups it is possible to specify the category of the group as the first argument: either GrpMat (a reflection group), GrpFPCox, GrpFP (a finitely presented group), GrpPermCox or GrpPerm (a permutation group). If the category is not supplied, 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.

CoxeterGroup(grpcat, N): Cat, MonStgElt -> grpcat

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.

CoxeterGroup(N): MonStgElt -> Grp: The finite Coxeter group of type GrpPermCox or the affine Coxeter group of type GrpFPCox with Cartan name given by the string \(N\) (see Section Finite and Affine Coxeter Groups).

IrreducibleCoxeterGroup(grpcat, X, n): Cat, MonStgElt, RngIntElt -> grpcat
IrreducibleCoxeterGroup(X, n): MonStgElt, RngIntElt -> .: The finite or affine irreducible Coxeter group of type grpcat with Cartan name \(X_n\), or \(I_2(n)\) if \(X=\)"I" (see Section Finite and Affine Coxeter Groups).

Example: Constructing a Coxeter Group by Name (ex-fcfb78)

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> 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)

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Note that if the category is not supplied, a permutation group of type GrpPermCox is returned for finite groups and a finitely presented group of type GrpFPCox is returned for infinite groups.

CoxeterGroup(grpcat, M): Cat, AlgMatElt -> grpcat: The Coxeter group of type grpcat with Coxeter or Cartan matrix \(M\) (see Chapter Coxeter Systems).

CoxeterGroup(grpcat, G): Cat, GrphUnd -> grpcat: The Coxeter group of type grpcat with Coxeter graph \(G\) (see Chapter Coxeter Systems).

CoxeterGroup(grpcat, D): Cat, GrphDir -> grpcat: The Coxeter group of type grpcat with Dynkin digraph \(D\) (see Chapter Coxeter Systems).

CoxeterGroup(M): AlgMatElt -> Grp: The Coxeter group of type GrpPermCox or GrpFPCox with Coxeter or Cartan matrix \(M\) (see Chapter Coxeter Systems).

CoxeterGroup(G): GrphUnd -> Grp: The Coxeter group of type GrpPermCox or GrpFPCox with Coxeter graph \(G\) (see Chapter Coxeter Systems).

CoxeterGroup(D): GrphDir -> Grp: The Coxeter group of type GrpPermCox or GrpFPCox with Dynkin digraph \(D\) (see Chapter Coxeter Systems).

Example: Coxeter Group from a Coxeter Matrix (ex-2b7d19)

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> 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

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CoxeterGroup(grpcat, R): Cat, RootStr -> grpcat: The finite Coxeter group of type grpcat with root system or root datum \(R\) (see Chapters Root Systems and Root Data).

CoxeterGroup(R): RootStr -> GrpPermCox: The finite Coxeter group of type GrpPermCox with root system or root datum \(R\) (see Chapters Root Systems and Root Data).

CoxeterGroup(A, B): Mtrx, Mtrx -> GrpPermCox

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:

\(A\) 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;
the number of columns must be at least the number of rows; and
\(AB^t\) must be the Cartan matrix of a Coxeter group.

CoxeterGroup(grpcat, A, B): Cat, Mtrx, Mtrx -> grpcat: The Coxeter group of type grpcat where grpcat is GrpMat, GrpPermCox, GrpPerm, or GrpFPCox. As above, the rows of \(A\) are the simple roots and the rows of \(B\) are the simple coroots.

Example: Coxeter Group from a Root Datum (ex-af5154)

> 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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Constructing Coxeter Groups#

Constructing Coxeter Groups