97 Hecke cosets

``Hecke cosets" are 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.

Subsections

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

97.1 Hecke for Coxeter cosets

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:

returns the untwisted Hecke algebra corresponding to the Hecke coset.

CoxeterCoset:

returns the Coxeter coset corresponding to the Hecke coset.

CoxeterGroup:

returns the untwisted Coxeter group corresponding to the Hecke coset.

Print:

prints the Hecke coset in a form which can be read back into GAP3.

CharTable:

returns the character table of the Hecke coset.

Basis(H,"T"):

the T basis.

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gap3-jm
19 Feb 2018