``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,
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
Previous Up Next