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.

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.

gap3-jm
11 Mar 2019