[GAP Forum] Class structure identification of symmetric groups

[Frank Lübeck]

On Thu, Jan 12, 2006 at 07:56:03AM +0000, raghunathan wrote:
> Hello GAP forum,
>         Is there a way to find the different classes of permutations 
> corresponding to different columns of the character table of a 
> symmetric group? 
> For ex., 
>         The character table of S3 is displayed by GAP as
>      2  1  1  .
>      3  1  .  1
> 
>        1a 2a 3a
>     2P 1a 1a 3a
>     3P 1a 2a 1a
> 
> X.1     1 -1  1
> X.2     2  . -1
> X.3     1  1  1
>           Is there a way to relate the symbols displayed just above the table
> i.e., 1a 2a 3a etc., to the different classes (1)(2)(3),(1)(2,3) &(1,2,3) 
> of SymmetricGroup(3)?

Dear Raghunathan, dear Forum,

In general it can be difficult to identify the conjugacy classes of a given
group in GAP with the columns of its abstract character table.

But for symmetric groups GAP can compute the character table, using the
labeling of conjugacy classes and irreducible characters by partitions:

gap> t := CharacterTable("Symmetric", 7); 
CharacterTable( "Sym(7)" )
gap> ClassParameters(t);
[ [ 1, [ 1, 1, 1, 1, 1, 1, 1 ] ], [ 1, [ 2, 1, 1, 1, 1, 1 ] ], 
  [ 1, [ 2, 2, 1, 1, 1 ] ], [ 1, [ 2, 2, 2, 1 ] ], [ 1, [ 3, 1, 1, 1, 1 ] ], 
  [ 1, [ 3, 2, 1, 1 ] ], [ 1, [ 3, 2, 2 ] ], [ 1, [ 3, 3, 1 ] ], 
  [ 1, [ 4, 1, 1, 1 ] ], [ 1, [ 4, 2, 1 ] ], [ 1, [ 4, 3 ] ], 
  [ 1, [ 5, 1, 1 ] ], [ 1, [ 5, 2 ] ], [ 1, [ 6, 1 ] ], [ 1, [ 7 ] ] ]

The i-th entry of ClassParameters(t) describes the class of the i-th column
of t: its second component gives the cycle type of the elements in this class.

There is a similar CharacterParameters(t).

Remark: If G is a group in GAP then 'CharacterTable(G);' returns a table
which can be asked for 'IdentificationOfConjugacyClasses', e.g.:

gap> G := SymmetricGroup(20);
Sym( [ 1 .. 20 ] )
gap> t := CharacterTable(G);
CharacterTable( Sym( [ 1 .. 20 ] ) )
gap> IdentificationOfConjugacyClasses(t);
[ 1 .. 627 ]

But for bigger G GAP may not be able to compute the character table.
Nevertheless, in this particular case of symmetric groups, GAP knows that 
it can use the efficient function mentioned above to compute the table.

With best regards,

  Frank Luebeck


-- 
///  Dr. Frank Lübeck, Lehrstuhl D für Mathematik, Templergraben 64,  ///
\\\                    52062 Aachen, Germany                          \\\
///  E-mail: Frank.Luebeck at Math.RWTH-Aachen.De                        ///
\\\  WWW:    http://www.math.rwth-aachen.de/~Frank.Luebeck/           \\\