See Chapter FINITELY PRESENTED GROUPS for general functions for finitely presented groups, or Chapter PERMUTATION GROUPS for general functions for permutation groups.
Returns true if, and only if, W1 and W2 are isomorphic as abstract groups. This is only implemented for permutation Coxeter groups.
Tests if W1 and W2 are isomorphic as Coxeter systems. If true, a sequence giving the permutation of the generators which takes W1 to W2 is also returned.
Returns true if and only if the crystallographic Coxeter groups W1 and W2 have Cartan equivalent Cartan matrices. This only makes sense for permutation Coxeter groups.
> W1 := CoxeterGroup(GrpFPCox, "B4"); > W2 := CoxeterGroup(GrpFPCox, "C4"); > IsCoxeterIsomorphic(W1, W2); true [ 1, 2, 3, 4 ]An example of abstractly isomorphic Coxeter groups whose Coxeter systems not isomorphic:
> W1 := CoxeterGroup("G2");
> W2 := CoxeterGroup("A1A2");
> IsIsomorphic(W1, W2);
true
> IsCoxeterIsomorphic(W1, W2);
false
An example of Coxeter isomorphic groups which are not Cartan equivalent:
> W1 := CoxeterGroup("B3");
> W2 := CoxeterGroup("C3");
> IsIsomorphic(W1, W2);
true
> IsCoxeterIsomorphic(W1, W2);
true [ 1, 2, 3 ]
> IsCartanEquivalent(W1, W2);
false
The underlying root system of the permutation Coxeter groupW.
The root datum of the permutation Coxeter groupW. If W does not have a root datum, an error is flagged.
> W := CoxeterGroup("C5");
> RootSystem(W);
Root system of type C5
> RootDatum(W);
Root datum of type C5
>
> W := CoxeterGroup("H4");
> RootSystem(W);
Root system of type H4
> RootDatum(W);
Error: This group does not have a root datum
The Cartan name of the finite or affine Coxeter group W (Section Finite and Affine Coxeter Groups).
Print the Coxeter diagram of the finite or affine Coxeter group W (Section Finite and Affine Coxeter Groups).
Print the Dynkin diagramof the permutation Coxeter group W. If W is not crystallographic, an error is flagged.
> W := CoxeterGroup("F4");
> CartanName(W);
F4
> DynkinDiagram(W);
F4 1 - 2 =>= 3 - 4
> CoxeterDiagram(W);
F4 1 - 2 === 3 - 4
The Coxeter matrix of the Coxeter group W.
The Coxeter graph of the Coxeter group W.
The Cartan matrix of the permutation Coxeter group W.
The Dynkin digraph of the permutation Coxeter group W.
The rank of the Coxeter group W.
The number of positive roots of the Coxeter group W.
The dimension of the permutation Coxeter group W, ie. the dimension of the root space.
> R := StandardRootSystem("A", 4);
> W := CoxeterGroup(R);
> Rank(W);
4
> Dimension(W);
5
The conjugacy classes of the finite Coxeter group W. This uses the algorithm of [GP00].
The fundamental groupof the permutation Coxeter group W. The roots and coroots of W must have integral components.
The isogeny groupof the permutation Coxeter group W. The roots and coroots of W must have integral components.
The coisogeny groupof the permutation Coxeter group W. The roots and coroots of W must have integral components.
The degrees of the basic invariant polynomials of the Coxeter group W. These are computed using the table in [Car72, page 155].
The basic codegrees of the Coxeter group W. These are computed using the algorithm in [LT09].
> W := CoxeterGroup("E6");
> degs := BasicDegrees(W);
> degs;
[ 2, 5, 6, 8, 9, 12 ]
> &*degs eq #W;
true
> &+degs eq NumPosRoots(W) + Rank(W);
true
If Coxeter group element x is less than or equal to y in the Bruhat order [Deo77]. Suppose x is an element of the Coxeter group W. The Bruhat order is the partial order generated by the relations: x ≤xw if l(x) < l(xw), and xw ≤x if l(xw) < l(x), for x∈W and w a reflection. If l(xw) = l(x) + 1, then x is called a Bruhat descendant of xw. The algorithm used is a straightforward recursive procedure.
z: GrpPermElt Default:
Let x be an element of the Coxeter group W, then the returned set S contains the Bruhat descendants of x. If l(yw) = l(y) + 1, then y is called a Bruhat descendant of yw. If the optional parameter z is set, only those descendants y with z ≤y are returned. Algorithm: For each fundamental reflection in x it is tested whether leaving it out decreases the length of x by exactly 1. If so, it is included in the result. In particular, this algorithm does not use BruhatLessOrEqual.
z: GrpPermElt Default:
Let X consist of elements of the Coxeter group W, then the returned set S contains the Bruhat descendants of every element of X.If the optional parameter z is set, only those w are returned for which z ≤w in the Bruhat ordering.
> R := RootDatum("D4" : Isogeny := "SC");
> W := CoxeterGroup(GrpPermCox, R);
> Wfp,phi := CoxeterGroup(GrpFPCox, W);
> x := W.1*W.3*W.2*W.4*W.2*W.2*W.2*W.1;
> Eltseq(phi(x));
[ 1, 3, 2, 4, 2, 1 ]
> S := BruhatDescendants(x);
> { Eltseq(phi(w)) : w in S };
{
[ 1, 3, 2, 4, 2 ],
[ 3, 2, 4, 2, 1 ],
[ 1, 2, 4, 2, 1 ],
[ 1, 3, 2, 1, 4 ],
[ 1, 3, 4, 2, 1 ]
}