81.5 Operations for Hecke elements of the $T$ basis

All examples below are with CHEVIE.PrintHecke="".

Hecke( a )

returns the Hecke algebra of which a is an element.

a * b

The multiplication of two elements given in the T basis of the same Iwahori-Hecke algebra is defined, returning a Hecke element expressed in the T basis.

    gap> q := X( Rationals );; q.name := "q";;
    gap> H := Hecke( CoxeterGroup( "A", 2 ), q );
    Hecke(CoxeterGroup("A", 2),[ q, q ],[  ])
    gap> T := Basis( H, "T" );
    function ( arg ) ... end
    gap> ( T() + T( 1 ) ) * ( T() + T( 2 ) );
    T()+T(1)+T(2)+T(1,2)
    gap> T( 1 ) * T( 1 );
    qT()+(q-1)T(1)
    gap> T( 1, 1 ); # the same
    qT()+(q-1)T(1) 

a ^ i

A element of the T basis with a coefficient whose inverse is still a Laurent polynomial in q can be raised to an integral, positive or negative, power, returning another element of the algebra. An arbitrary element of the algebra can only be raised to a positive power.

    gap> ( q * T( 1, 2 ) ) ^ -1;
    (q^-1-2q^-2+q^-3)T()+(-q^-2+q^-3)T(1)+(-q^-2+q^-3)T(2)+q^-3T(2,1)
    gap> ( T( 1 ) + T( 2 ) ) ^ -1;
    Error, negative exponent implemented only for single T_w in
    <rec1> ^ <rec2> called from
    main loop
    brk> 
    gap> ( T( 1 ) + T( 2 ) ) ^ 2;
    2qT()+(q-1)T(1)+(q-1)T(2)+T(1,2)+T(2,1) 

a / b

This is equivalent to a* b^{-1}.

a + b

a - b

Elements of the algebra expressed in the T basis can be added or subtracted, giving other elements of the algebra.

    gap> T( 1 ) + T();
    T()+T(1)
    gap> T( 1 ) - T( 1 );
    0 

Normalize( a )

normalizes the element a; in particular, terms with zero coefficient are removed.

    gap> h := T( [ PermCoxeterWord( CoxeterGroup( H ), [ 1 ] ),() ], 
    >                                                     [ 0, q^100 ] );
    q^100T()+0T(1)
    gap> Normalize( h );
    gap> h;
    q^100T() 

Print( a )

prints the element a, using the form initialized in CHEVIE.PrintHecke.

String( a )

provides a string containing the same result that is printed with Print.

Coefficient( a, w )

Returns the coefficient of the Hecke element a on the basis element T_w. Here w can be given either as a Coxeter word or as a permutation.

AlphaInvolution( a )

This implements the involution on the algebra defined by T_wmapsto T_{w^{-1}}.

    gap> AlphaInvolution( T( 1, 2 ) );
    T(2,1) 

BetaInvolution( a )

This is only defined if all the parameters of the Iwahori-Hecke algebra are equal, and they are either equal to 1 or all sqrtParameters are bound. If v is the square root of the first parameter, it implements the involution on the algebra defined by vmapsto v^{-1} and T_wmapsto v^{-l(w_0)}T_{w_0w}.

AltInvolution( a )

This is only defined if all the parameters of the Iwahori-Hecke algebra are equal, and they are either equal to 1 or all sqrtParameters are bound. If v is the square root of the first parameter, it implements the involution on the algebra defined by vmapsto -v^{-1} and T_wmapsto(-v^{-2})^{l(w)}T_w. Essentially it corresponds to tensoring with the sign representation.

Frobenius(WF)( a )

The Frobenius of a Coxeter Coset associated to CoxeterGroup(Hecke(a)) can be applied to a. For more details see chapter Coxeter cosets.

These functions require the package "chevie" (see RequirePackage). Previous Up Top Next
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