# 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.

## 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
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GAP 3.4.4
April 1997