84.3 Functions on Coxeter cosets

CoxeterGroup( WF ):

returns the Coxeter group of which WF is a coset.

Quite a few functions defined for domains, permutation groups or Coxeter groups have been implemented to work with Coxeter cosets.

Elements, Random, Representative, Size, in:

these functions use the corresponding functions for CoxeterGroup( WF ).

ConjugacyClasses( WF ):

returns the conjugacy classes of the Coxeter coset WF (see also the introduction of this Chapter). Let W be CoxeterGroup(WF). Then the classes are defined to be the W-orbits on W F_0, where W acts by conjugation (they coincide with the W F_0-orbits, W F_0 acting by the conjugation); by the translation wmapsto wphi^{-1} they are sent to the phi-conjugacy classes of W.

PositionClass( WF , x ):

for any element x in WF this returns the number i such that x is an element of ConjugacyClasses(WF)[i] (to work fast, the classification of Coxeter groups is used).

FusionConjugacyClasses( WF1, WF ):

works in the same way as for groups. See the section CoxeterSubCoset.

Print( WF ):

if WF.name is bound then this is printed, else this function prints the coset in a form which can be input back into GAP.

InductionTable( HF, WF ):

works in the same way as for groups. It gives the induction table from the Coxeter subcoset HF to the Coxeter coset WF. If H wF_0 is a Coxeter subcoset of W F_0, restriction of characters is defined as restriction of functions from W F_0 to H wF_0, and induction as the adjoint map for the natural scalar product < f, g> ={1over #W}sum_{v in W} f(v F_0)overline g(v F_0).
If the Coxeter coset WF represents the reductive group bG, and HF corresponds to a Levi subgroup bL, then the induction from HF to WF describes the Lusztig induction of uniform unipotent almost characters from bL to bG.

Harish-Chandra induction in the basis of almost characters:

    gap> WF := CoxeterCoset( CoxeterGroup( "A", 4 ), (1,4)(2,3) ); 
    CoxeterCoset(CoxeterGroup("A", 4), (1,4)(2,3))
    gap> Display( InductionTable( CoxeterSubCoset( WF, [ 2, 3 ] ), WF ) );
          

 111 21 3
    ________________
    11111 

   1  . .
    2111  

   .  1 .
    221   

   1  . .
    311   

   1  . 1
    32    

   .  . 1
    41    

   .  1 .
    5     

Lusztig induction from a diagonal Levi:

    gap> HF := CoxeterSubCoset( WF, [1, 2], 
    >                LongestCoxeterElement( CoxeterGroup( WF ) ) );;
    gap> Display( InductionTable( HF, WF ) );

          

 111 21  3
    _________________
    11111 

  -1  .  .
    2111  

  -2 -1  .
    221   

  -1 -2  .
    311   

   1  2 -1
    32    

   . -2  1
    41    

   .  1 -2
    5     

A descent of scalars:

    gap> W := CoxeterCoset( CoxeterGroup( "A", 2, "A", 2 ), (1,3)(2,4) ); 
    CoxeterCoset(CoxeterGroup("A", 2, "A", 2), (1,3)(2,4))
    gap> Display( InductionTable( CoxeterSubCoset( W, [ 1, 3 ] ), W ) );
        

 11 2
    __________
    111 

  1 .
    21  

  1 1
    3   

CartanName( WF ):

returns a string which describes the isomorphism type of the group Wrtimes < F> , associated to WF, as described in the introduction of this Chapter. An orbit of phi=WF.phi on the components is put in brackets if of length k greater than 1, and is preceded by the order of phi^k on it, if this is not 1. For example "2(A2xA2)" denotes 2 components of type A_2 permuted by F_0, and such that phi^2 induces the non-trivial diagram automorphism on any of them, while 3D4 denotes an orbit of length 1 on which phi is of order 3.

    gap> W := CoxeterCoset( CoxeterGroup( "A", 2, "G", 2, "A", 2 ), 
    >                                                       (1,5,2,6) );
    CoxeterCoset(CoxeterGroup("A", 2, "G", 2, "A", 2), (1,5,2,6))
    gap> CartanName( W );
    "2(A2xA2)xG2" 

PrintDynkinDiagram( WF ):

this is a purely descriptive routine (as was already the case for finite Coxeter groups themselves). It prints the Dynkin diagram of CoxeterGroup(WF) together with the information how WF.phi acts on it.

    gap> W := CoxeterCoset( CoxeterGroup( "A", 2, "A", 2 ), (1,3,2,4) );
    CoxeterCoset(CoxeterGroup("A", 2, "A", 2), (1,3,2,4))
    gap> PrintDynkinDiagram( W );
    phi permutes the next 2 components
    phi^2 acts as (1,2) on the component below
    A2    1 - 2
    A2    3 - 4 

ChevieClassInfo( WF ), see the explicit description in ChevieClassInfo for Coxeter cosets.

ChevieCharInfo:

This function returns additional information on the irreducible characters, see ChevieCharInfo for more details.

CharParams( WF ) CharName( WF )

Note that some functions for elements of a Coxeter group work naturally for elements of a Coxeter coset: CoxeterWord, PermCoxeterWord, CoxeterLength, ReducedInCoxeterCoset, LeftDescentSet, RightDescentSet, etcldots

Previous Up Top Next
Index