85 Hecke cosets

``Hecke cosets" are Hphi where H is a Hecke algebra of some Coxeter group W on which the reduced element phi acts by phi(T_w)=T_{phi(w)}. This corresponds to the action of the Frobenius automorphism on the commuting algebra of the induced of the trivial representation from the rational points of some F-stable Borel subgroup to {bf G}^F.

    gap> W := CoxeterGroup( "A", 2 );;
    gap> q := X( Rationals );; q.name := "q";;
    gap> HF := Hecke( CoxeterCoset( W, (1,2) ), q^2, q );
    Hecke(CoxeterCoset(CoxeterGroup("A", 2), (1,2)),[ q^2, q^2 ],[ q, q ])
    gap> Display( CharTable( HF ) );
    H(2A2)

         2     1   1   .
         3     1   .   1

             111  21   3
        2P   111 111   3
        3P   111  21 111

    111       -1   1  -1
    21     -2q^3   0   q
    3        q^6   1 q^2 

We do not yet have a satisfying theory of character tables for these cosets (the equivalent of HeckeClassPolynomials has not yet been proven to exist). We hope that future releases of CHEVIE will contain better versions of such character tables.

Subsections

1. Hecke for Coxeter cosets
2. Operations and functions for Hecke cosets