Operations on Elements

See Chapter [DRAFT] FINITELY PRESENTED GROUPS for general functions for finitely presented groups or Chapter ChapGrpPerm for general functions for permutation groups.

Unlike groups of type GrpFP, elements of a group of type are always converted into the normal form of Section The Normal Form for Words.

Example: Word Arithmetic

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
%%a> assert $1;

\# w: GrpFPCoxElt → RngIntElt

Length(w): GrpFPCoxElt → RngIntElt

Length(W, w): GrpPermCox, GrpPermElt → RngIntElt

CoxeterLength(w): GrpFPCoxElt → RngIntElt

CoxeterLength(W, w): GrpPermCox, GrpPermElt → RngIntElt

The length of \(w\) 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): GrpFPCox → SeqEnum

LongestElement(W): GrpPermCox → GrpPermElt

The unique longest element of the Coxeter group \(W\).

CoxeterElement(W): GrpFPCox → SeqEnum

CoxeterElement(W): GrpPermCox → GrpPermElt

The Coxeter element of the Coxeter group \(W\), ie. the product of the generators of \(W\).

CoxeterNumber(W): GrpFPCox → SeqEnum

CoxeterNumber(W): GrpPermCox → GrpPermElt

The Coxeter number of the irreducible Coxeter group \(W\) (see :raw-latex:`\cite[page 20]{Carter-big}`).

Example: Longest Coxeter Elements

The Coxeter number can be described in a variety of ways.

> 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
%%a> assert $1 eq 120;
> Length(W, CoxeterElement(W));
8
%%a> assert $1 eq 8;
> W := CoxeterGroup("D5");
> CoxeterNumber(W) eq Order(CoxeterElement(W));
true
%%a> assert $1;
> CoxeterNumber(W) eq #Roots(W) / Rank(W);
true
%%a> assert $1;
> R := RootDatum(W);
> CoxeterNumber(W) eq &+Eltseq(HighestRoot(R)) + 1;
true
%%a> assert $1;

LeftDescentSet(W, w): GrpFPCox, GrpFPCoxElt → $\{\}$

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 \(s_rw\) is less than that of the element \(w\).

RightDescentSet(W, w): GrpFPCox, GrpFPCoxElt → $\{\}$

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 \(ws_r\) is less than that of the element \(w\).

Example: Descent Sets

> W := CoxeterGroup("A5");
> x := W.1*W.2*W.4*W.5;
> LeftDescentSet(W, x);
{ 1, 4 }
> RightDescentSet(W, x);
{ 2, 5 }