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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.
Arithmetic with 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
Length(w) : GrpFPCoxElt -> RngIntElt
Length(W, w) : GrpPermCox, GrpPermElt -> RngIntElt
CoxeterLength(w) : GrpFPCoxElt -> RngIntElt
CoxeterLength(W, w) : GrpPermCox, GrpPermElt -> RngIntElt
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.
LongestElement(W) : GrpPermCox -> GrpPermElt
The unique longest elementof the Coxeter group W.
CoxeterElement(W) : GrpPermCox -> GrpPermElt
The Coxeter elementof the Coxeter group W, ie. the product of the
generators of W.
CoxeterNumber(W) : GrpPermCox -> GrpPermElt
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
LeftDescentSet(W, w) : GrpPermCox, GrpPermElt -> ()
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.
RightDescentSet(W, w) : GrpPermCox, GrpPermElt -> ()
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.
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