Constructing Coxeter Groups

CoxeterGroup( GrpFPCox, M ) : Cat, AlgMatElt -> GrpFPCox

CoxeterGroup( GrpFPCox, G ) : Cat, GrphUnd -> GrpFPCox

CoxeterGroup( GrpFPCox, C ) : Cat, AlgMatElt -> GrpFPCox

CoxeterGroup( GrpFPCox, D ) : Cat, GrphDir -> GrpFPCox

The Coxeter group with Coxeter matrix M or Coxeter graph G, Cartan matrix C, or Dynkin digraph D (see Chapter COXETER SYSTEMS).

Example GrpFPCox_ContructFromMatrix (H83E1)

> 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()

> C := CartanMatrix( "G2" );                                                   
> W := CoxeterGroup( GrpFPCox, C );

> CartanMatrix( W );

>> CartanMatrix( W );
               ^
Runtime error in 'CartanMatrix': Bad argument types
Argument types given: GrpFPCox    

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

The finite or affine Coxeter group with Cartan name N (see Section Finite and Affine Coxeter Groups).

Example GrpFPCox_ConstructByName (H83E2)

> 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( GrpFPCox, R ) : Cat, RootSys -> GrpFPCox

CoxeterGroup( GrpFPCox, R ) : Cat, RootDtm -> GrpFPCox

The finite Coxeter group with root system or root datum R (see Chapters ROOT SYSTEMS and ROOT DATA).

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' (see Chapter COXETER GROUPS AS PERMUTATION GROUPS).

CoxeterGroup( GrpFPCox, W ) : Cat, GrpMat -> GrpFPCox

The finitely presented Coxeter group W' isomorphic to the real reflection group W (see Chapter REFLECTION GROUPS).