``Hecke cosets" are Hφ where H is a Hecke algebra of some Coxeter group W on which the reduced element φ acts by φ(Tw)=Tφ(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 GF.
gap> W := CoxeterGroup( "A", 2 );; gap> q := X( Rationals );; q.name := "q";; gap> HF := Hecke( CoxeterCoset( W, (1,2) ), q^2, q ); Hecke(2A2,q^2,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 . q 3 q^6 1 q^2
Thanks to the work of Xuhua He and Sian Nie, HeckeClassPolynomials
also
make sense for these cosets. This is used to compute such character tables.
Hecke( WF, H )
Hecke( WF, params )
Construct a Hecke coset a Coxeter coset WF and an Hecke algebra
associated to the CoxeterGroup of WF. The second form is equivalent to
Hecke( WF, Hecke(CoxeterGroup(WF), params))
.
97.2 Operations and functions for Hecke cosets
Hecke
:CoxeterCoset
:CoxeterGroup
:
Print
:CharTable
:Basis(H,"T")
:T
basis.
gap3-jm