82.5 HeckeClassPolynomials

HeckeClassPolynomials( h [, reps] )

returns the class polynomials of the Hecke element h of the Hecke algebra H with respect to representatives reps of minimal length in the conjugacy classes of the Coxeter group Group(H).

If absent, reps is taken as CoxeterConjugacyClasses(Group(H)). These polynomials have the following property. Given the class polynomials p corresponding to h and the matrix X of the values of the irreducible characters of the Iwahori-Hecke algebra on T_w (for w in reps), then the product X*p is the list of values of the irreducible characters on the element h of the Iwahori-Hecke algebra.

    gap> u := X( Rationals );; u.name := "u";;
    gap> W := CoxeterGroup( "A", 3 );
    CoxeterGroup("A", 3)
    gap> H := Hecke( W, u );; 
    gap> h := Basis( H, "T" )( LongestCoxeterElement( W ) );
    T(1,2,1,3,2,1)
    gap> cp := HeckeClassPolynomials( h );
    [ 0*u^0, 0*u^0, u^2, u^3 - 2*u^2 + u, u^3 - u^2 + u - 1 ]
    gap> CharTable( H ).irreducibles * cp;
    [ u^0, -u^2, 2*u^3, -u^4, u^6 ] 

So, the entries in this list are the values of the irreducible characters on the basis element corresponding to the longest element in the Coxeter group.

The class polynomials were introduced in GP93.

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

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