Coxeter groups can be described in a standard manner as finitely presented groups. See Chapter FINITELY PRESENTED GROUPS for more details on finitely presented groups.
Construct the Coxeter group of Cartan type t as a finitely presented group, given by the standard Coxeter presentation. The Cartan type t is passed to this function as a string; we refer to Chapter ROOT DATA FOR LIE THEORY for details.
To construct a finitely presented group, given in standard Coxeter presentation, from other data, e.g. a Cartan matrix or a root datum, it is recommended to create a Coxeter group in category GrpCox first, and then create an FP Coxeter group as described in Subsection Conversion.
> F := CoxeterGroup(GrpFP, "F4");
> F;
Finitely presented group F on 4 generators
Relations
(F.2 * F.3)^2 = (F.3 * F.2)^2
F.1^2 = Id(F)
F.1 * F.3 = F.3 * F.1
F.2 * F.4 = F.4 * F.2
F.1 * F.2 * F.1 = F.2 * F.1 * F.2
F.2^2 = Id(F)
F.3^2 = Id(F)
F.3 * F.4 * F.3 = F.4 * F.3 * F.4
F.4^2 = Id(F)
F.1 * F.4 = F.4 * F.1
Local: BoolElt Default: false
The FP Coxeter group F of the finite Coxeter group W, and the isomorphism W to F. The first argument must be the category GrpFP If the optional parameter Local is present, F is the appropriate subgroup of the FP version of the overgroup of W.
The Coxeter group W of the FP Coxeter group F, and the isomorphism F to W.
> W := CoxeterGroup( "C5" );
> F<[s]>, h := CoxeterGroup( GrpFP, W );
> h( W.1*W.2 );
s[1] * s[2]
> hinv := Inverse( h );
> hinv( F![1,2,3,4,5,1,2,3,4] );
(1, 34, 46, 42, 28, 26, 9, 21, 17, 3)(2, 39, 16, 24, 29, 27, 14, 41, 49,
4)(5, 19, 25, 38, 48, 30, 44, 50, 13, 23)(6, 45, 32, 11, 33, 31, 20, 7,
36, 8)(10, 43, 40, 37, 47, 35, 18, 15, 12, 22)
The unique longest element in the FP Coxeter group F.
The Coxeter element in the FP Coxeter group F, ie. the product of the generators.
An additive order on the positive roots of F, ie. a sequence containing the numbers 1, ..., N in some order so that alpha_r + alpha_s=alpha_t implies t is between r and s. This is computed using the techniques of [Pap94]
> W := CoxeterGroup( "F4" ); > F<[s]> := CoxeterGroup( GrpFP, W ); > LongestElement( F ); s[1] * s[2] * s[1] * s[3] * s[2] * s[1] * s[3] * s[2] * s[3] * s[4] * s[3] * s[2] * s[1] * s[3] * s[2] * s[3] * s[4] * s[3] * s[2] * s[1] * s[3] * s[2] * s[3] * s[4] > CoxeterElement( F ); s[1] * s[2] * s[3] * s[4] > AdditiveOrder( F ); [ 4, 7, 16, 10, 18, 20, 12, 21, 13, 22, 15, 23, 24, 17, 19, 3, 9, 6, 11, 14, 8, 2, 5, 1 ]
The product w_1w_2 in the FP Coxeter group F. If w_1 and w_2 are reduced words, then the output will also be reduced. The words w_1 and w_2 may be given either as elements of F or as sequences of integers.
A reduced word for w in the FP Coxeter group F. The word w may be given either as an element of F or as a sequence of integers.
The inverse of w in the FP Coxeter group F. If w is a reduced word, then the output will also be reduced. The word w may be given either as an element of F or as a sequence of integers.
True if w_1 and w_2 are equal in the FP Coxeter group F. The words w_1 and w_2 may be given either as elements of F or as sequences of integers.
> W := CoxeterGroup( "G2" ); > F := CoxeterGroup( GrpFP, W ); > w1 := Reduce( F, [1,2,1,2,1] ); > w1; s[1] * s[2] * s[1] * s[2] * s[1] > w2 := Reduce( F, [1,2,2,1,2,1] ); > w2; s[2] * s[1] > WordProduct( F, w1, Inverse( F, w2 ) ); s[1] * s[2] * s[1] > IsEqual( F, [1,2,1,2,1,2], [2,1,2,1,2,1] ); true
The action of the FP Coxeter group F on the indices of the (co)roots.
basis: MonStgElt Default: "standard"
The action of the FP Coxeter group F on the indices of the (co)root space.
> W := CoxeterGroup( "I2(5)" ); > F<[s]> := CoxeterGroup( GrpFP, W ); > w := s[1] * s[2]; > act := Action( F ); > act(1,w); 8 > act(8,w); 9The function RootAction gives an action on elements of the root space, including the roots themselves.
> W := CoxeterGroup( "I2(5)" ); > F<[s]> := CoxeterGroup( GrpFP, W ); > w := s[1] * s[2]; > act := RootAction( F ); > act([1,1],w); (-z^3 - z^2 - 1 0)
The braid group B of the Coxeter group W as a finitely presented group, together with the natural map W to B.
The braid group B of the FP Coxeter group F as a finitely presented group, together with the projection B to F. The braid group is the group with the same generators and braid relations as F, but with no order relations.
Returns the pure braid group of the Coxeter group W, i.e. the kernel of the epimorphism from the Braid group of W to W, as a finitely presented group.
Returns the pure braid group of the FP Coxeter group F, i.e. the kernel of the epimorphism from the Braid group of F to F, as a finitely presented group.
> W := CoxeterGroup( "B3" );
> F := CoxeterGroup( GrpFP, W );
> F;
Finitely presented group F on 3 generators
Relations
(F.2 * F.3)^2 = (F.3 * F.2)^2
F.1^2 = Id(F)
F.1 * F.3 = F.3 * F.1
F.1 * F.2 * F.1 = F.2 * F.1 * F.2
F.2^2 = Id(F)
F.3^2 = Id(F)
> B := BraidGroup( W );
> B;
Finitely presented group B on 3 generators
Relations
(B.2 * B.3)^2 = (B.3 * B.2)^2
B.1 * B.3 = B.3 * B.1
B.1 * B.2 * B.1 = B.2 * B.1 * B.2
> P := PureBraidGroup( W );
> P;
Finitely presented group P on 3 generators
Generators as words
P.1 = B.1^2
P.2 = B.2^2
P.3 = B.3^2