Operations on Words

See Chapter FINITELY PRESENTED GROUPS for general functions for finitely presented groups.

W ! [i_1,...,i_m] : GrpFPCox, SeqEnum[RngIntElt] -> GrpFPCoxElt

The word s_(i_1) ... s_(i_m) converted to normal form.

Identity( W ) : GrpFPCox -> GrpFPCoxElt

Id( W ) : GrpFPCox -> GrpFPCoxElt

G ! 1 : GrpFPCox, RngIntElt -> GrpFPCoxElt

The identity element, represented as the empty word.

w1 eq w2 : GrpFPElt, GrpFPElt -> BoolElt

Returns true if, and only if, w_1 and w_2 are equal as Coxeter group elements, i.e. they are equal as words.

w1 * w2 : GrpFPCoxElt, GrpFPCoxElt -> GrpFPCoxElt

The product w_1w_2 in normal form.

w ^ n : GrpFPCoxElt, RngIntElt -> GrpFPCoxElt

The nth power of the word w in normal form, where n is an integer.

# w : GrpFPCoxElt -> RngIntElt

CoxeterLength( W, w ) : GrpFPCox,GrpFPCoxElt -> RngIntElt

The Coxeter length of w in the Coxeter group W.

Example GrpFPCox_WordArithmetic (H83E7)

> 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

LongestElement( W ) : GrpFPCox -> SeqEnum

The unique longest element in the finite Coxeter group W.

CoxeterElement( W ) : GrpFPCox -> SeqEnum

The Coxeter element in the Coxeter group W, ie. the product of the generators.

CoxeterNumber( W ) : GrpFPCox -> SeqEnum

The order of the Coxeter element in the Coxeter group W.

Example GrpFPCox_SpecialWords (H83E8)

> 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]

LeftDescentSet( W, w ) : GrpFPCox, GrpFPCoxElt -> {}

The set of indices r of simple roots of the finite 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 -> {}

The set of indices r of simple roots of the finite Coxeter group W such that the length of the product ws_r is less than that of the element w.

Example GrpFPCox_DescentSets (H83E9)

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