# 97 Hecke cosets

``Hecke cosets" are *Hφ* where *H* is a Hecke algebra of some
Coxeter group *W* on which the reduced element *φ* acts by
*φ(T*_{w})=T_{φ(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 ** 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(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

- Hecke for Coxeter cosets
- 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`))

.

`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

11 Mar 2019