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

85.1 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)).

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

85.2 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 GAP.

CharTable:

returns the character table of the Hecke coset.

These functions require the package "chevie" (see RequirePackage). Previous Up Next
Index

GAP 3.4.4
April 1997
======= GAP Manual: 85 Hecke cosets

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

85.1 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)).

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

85.2 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 GAP.

CharTable:

returns the character table of the Hecke coset.

These functions require the package "chevie" (see RequirePackage). Previous Up Next
Index

GAP 3.4.4
January 1997