86.1 InductionTable

InductionTable( W1, W )

InductionTable computes the decomposition of the induced characters from the subgroup W1 into irreducible characters of W. The rows correspond to the characters of the parent group, the columns to those of the subgroup. What is returned is actually a record with several fields: scalar contains the induction table proper, and there is a Display method. The other fields contain labeling information taken from the character tables of W1 and W when it exists.

    gap> W := Group( [ (1,2), (2,3), (3,4) ], () );
    Group( (1,2), (2,3), (3,4) )
    gap> H:=Subgroup( W, [ (1,2), (3,4) ] );
    Subgroup( Group( (1,2), (2,3), (3,4) ), [ (1,2), (3,4) ] )
    gap> W.name := "W";; H.name := "H";; # to avoid warnings
    gap> Display( InductionTable( H, W ) );
        

 X.1 X.2 X.3 X.4
    ______________________________
    X.1 

   1   .   .   .
    X.2 

   .   .   .   1
    X.3 

   1   .   .   1
    X.4 

   .   1   1   1
    X.5 

    gap> W := CoxeterGroup( "G", 2 );;
    gap> H := ReflectionSubgroup( W, [ 1, 4 ] );
    ReflectionSubgroup(CoxeterGroup("G", 2), [ 1, 4 ])
    gap> CartanName( H );
    "A1x~A1"
    gap> t := InductionTable( H, W );       
    InductionTable( ReflectionSubgroup(CoxeterGroup("G", 2), 
    [ 1, 4 ]), CoxeterGroup("G", 2))
    gap> Display( t );
                

 11,11 11,2 2,11 2,2
    __________________________________________
    phi_{1,0}   

     .    .    .   1
    phi_{1,6}   

     1    .    .   .
    phi_{1,3}'  

     .    1    .   .
    phi_{1,3}'' 

     .    .    1   .
    phi_{2,1}   

     .    1    1   .
    phi_{2,2}   

If one does not want to see the whole induction table, one can specify the characters of the subgroup and of the parent group by giving a second argument to Display. This second argument is a record with optional components charsGroup and charsSubgroup, to which one has to assign the lists of rows and columns which should be printed.

    gap> Display( t,rec( charsGroup := [5], charsSubgroup := [2,3] ) );
    Induction from A1x~A1 into G2
              

 11,2 2,11
    ______________________________
    phi_{2,1} 

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