79.2 Operations for braid elements

A braid element bb corresponding to the Coxeter group W is represented as a record with fields pw0, elm and operations which contains the operations described below. All examples below are with CHEVIE.PrintBraid="Deligne".

b1 * b2

The multiplication of two braid elements is defined, if they are braid elements of the same group, returning a braid element.

    gap> W := CoxeterGroup( "A", 2 );;
    gap> a := Braid( W )( [1] );
    1
    gap> b := Braid( W )( [2] );
    2
    gap> a * b;
    12 

b1 ^ i

A braid element can be raised to an integral, positive or negative, power, returning a braid element.

    gap> ( a * b ) ^ 4;
    w0^2.12
    gap> ( a * b ) ^ -1;
    w0^-1.2 

b1 ^ b2

This returns b_1^{-1}b_2b_1.

    gap> a ^ b;
    w0^-1.21.12 

b1 / b2

This returns b_1b_2^{-1}.

    gap> a / b;
    w0^-1.2.21 

CoxeterGroup( b )

This function returns the Coxeter group for which b is a braid element.

    gap> CoxeterGroup( a );
    CoxeterGroup("A", 2) 

String( b )

Print( b )

String returns a display form of the element b, and Print prints the result of String. The way elements are printed depends on the global variable CHEVIE.PrintBraid. If set to "GAP", the elements are printed in a form which can be read in back by the function Braid(W). If set to "Deligne" (resp. "Charney") the Deligne (resp. Charney) normal form (as explained in the introduction) is printed:

    gap> CHEVIE.PrintBraid := "GAP";;
    gap> ( a * b ) ^ -1;
    B([ 2 ],-1)
    gap> CHEVIE.PrintBraid := "Charney";;
    gap> ( a * b ) ^ -1;              
    (12)^-1.
    gap> CHEVIE.PrintBraid := "Deligne";;
    gap> ( a * b ) ^ -1;              
    w0^-1.2 

Frobenius( WF )( b )

The Frobenius of a Coxeter coset associated to CoxeterGroup(b) can be applied to b. See the Chapter on Coxeter cosets.

This function requires the package "chevie" (see RequirePackage).

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