See Chapter FINITELY PRESENTED GROUPS for general functions for finitely presented groups or Chapter PERMUTATION GROUPS for general functions for permutation groups.
Unlike groups of type GrpFP, elements of a group of type GrpFPCox are always converted into the normal form of Section The Normal Form for Words.
> W<[s]> := CoxeterGroup(GrpFPCox, "G2"); > w1 := W![2,1,2,1,2] ; > w1; s[2] * s[1] * s[2] * s[1] * s[2] > w2 := W![1,2,2,1,2,1]; > w2; s[2] * s[1] > w1 * w2; s[1] * s[2] * s[1] > W![1,2,1,2,1,2] eq W![2,1,2,1,2,1]; true
The lengthof w as an element of the Coxeter group W, ie. the number of positive roots of W which become negative under the action of w. The # operator does not work for permutation Coxeter group elements.
The unique longest elementof the Coxeter group W.
The Coxeter elementof the Coxeter group W, ie. the product of the generators of W.
The Coxeter numberof the irreducible Coxeter group W (see [Car93, page 20]).
> W<[s]> := CoxeterGroup(GrpFPCox, "F4");
> LongestElement(W);
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(W);
s[1] * s[2] * s[3] * s[4]
> W := CoxeterGroup("E8");
> Length(W, LongestElement(W));
120
> Length(W, CoxeterElement(W));
8
The Coxeter number can be described in a variety of ways.
> W := CoxeterGroup("D5");
> CoxeterNumber(W) eq Order(CoxeterElement(W));
true
> CoxeterNumber(W) eq #Roots(W) / Rank(W);
true
> R := RootDatum(W);
> CoxeterNumber(W) eq &+Eltseq(HighestRoot(R)) + 1;
true
The set of indices r of simple roots of the Coxeter group W such that the length of the product srw is less than that of the element w.
The set of indices r of simple roots of the Coxeter group W such that the length of the product wsr is less than that of the element w.
> W := CoxeterGroup("A5");
> x := W.1*W.2*W.4*W.5;
> LeftDescentSet(W, x);
{ 1, 4 }
> RightDescentSet(W, x);
{ 2, 5 }