GAP-Character table

Question

I the following link I have found the character table of $S_8$ which is computed with the program GAP.

http://groupprops.subwiki.org/wiki/Linear_representation_theory_of_symmetric_groups

But I don't quite understand it. For instance this row:

Character( CharacterTable( "Sym(8)" ), [ 28, -10, 4, -2, -4, 1, -1, 1, 1, -1, 2, 0, 2, -1, 0, -2, 0, 1, 1, -1, 0, 0 ] ),

First for which partition we get this row in the character table and second each column of this row associated to which conjugacy class?

I would be very thankful, if you please help me understanding these.

Answer 1

Here is a character table from GAP.

Here is how I made it:

gap> displaySym := function(n,file)
> local c,f;
> if IsString(file) then f := OutputTextFile(file,false);
>   SetPrintFormattingStatus(f,false);
> else f:= file; fi;
> c := CharacterTable("Symmetric",n);
> PrintTo(f, "\\documentclass[border=10]{standalone}\n");
> PrintTo(f, "\\begin{document}\n");
> PrintTo(f, "\\(\\begin{array}{l|",
>   ListWithIdenticalEntries(NrConjugacyClasses(c),'r'),"}\n");
> PrintTo(f, "Sym(",n,") & ", JoinStringsWithSeparator( List(CharacterParameters(c),
>   x -> Concatenation( List(Collected(x[2]),
>   y -> Concatenation( String(y[1]), "^{", String(y[2]), "}" ) ) ) )," & " ),
>   "\\\\ \\hline\n" );
> Perform( [1..NrConjugacyClasses(c)], function(k)
>   PrintTo( f, Concatenation( List( Collected( CharacterParameters(c)[k][2] ),
>   y -> Concatenation( String(y[1]), "^{", String(y[2]), "}" ) ) ), " & ",
>   JoinStringsWithSeparator( Irr(c)[k], " & " ), "\\\\\n" );
> end );
> PrintTo(f,"\\end{array}\\)\n");
> PrintTo(f,"\\end{document}\n");
> if IsString(file) then CloseStream(f); fi;
> end;;
gap> displaySym(8,"sym8.tex");;

Then convert to an image with:

$ xelatex sym8 >/dev/null 2>&1
$ convert -density 300 sym8.pdf sym8.png
$ pngcrush -e _c.png sym8.png