[GAP Forum] Class structure identification of symmetric groups

[raghunathan]

>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


Respected Dr.Frank Luebeck and GAP Forum,
          Thank you very much for ur valuable suggestion regarding class structures of symmetric groups which can be correlated to numerical partitions of the degree of the symmetric groups.But, for a wreath product group as simple as S2[S2] whose conjugacy classes' structures as given by GAP is as

gap> > [ ()^G, (3,4)^G, (1,2)(3,4)^G, (1,3)(2,4)^G, (1,3,2,4)^G ]

has two classes of permutations each with two cycles of length two (1,2)(3,4) and (1,3)(2,4). Also, the sequential commands, 

gap>s:=SymmetricGroup(2);;w:=WreathProduct(s,s);;t:=CharacterTable(w);;ClassParameters(t); 
         did not work.Is there a way to overcome this problem by relating the abstract symbols usually displayed above any character table given by GAP as 1a 2a 2b...etc., which in the case of S4 is as follows to the class structures?

     2  3  2  3  .  2
     3  1  .  .  1  .

       1a 2a 2b 3a 4a
    2P 1a 1a 1a 3a 2b
    3P 1a 2a 2b 1a 4a

X.1     1 -1  1  1 -1
X.2     3 -1 -1  .  1
X.3     2  .  2 -1  .
X.4     3  1 -1  . -1
X.5     1  1  1  1  1

Thanks,
Raghunathan,R.,
M.Sc.Chemistry,
Department of Chemistry,
Indian Institute of Technology Madras,
Chennai-36,
India.