[GAP Forum] Print the irreducible characters of the normal ordinary character table format.

[Hongyi Zhao]

On Sat, Apr 30, 2022 at 4:44 PM Bill Allombert
<Bill.Allombert at math.u-bordeaux.fr> wrote:
>
> On Sat, Apr 30, 2022 at 03:21:10PM +0800, Hongyi Zhao wrote:
> > Hi GAP team,
> >
> > I use the following code snippet to compute the irreducible characters
> > of a finitely presented group:
> >
> > So, I want to get a pretty printed result with the rows correspond to
> > irreducible representations, and the columns correspond to conjugacy
> > classes of group elements.
> >
> > Are there any clues to achieve this goal?
>
> Use Display(CharacterTable(g));
>
> gap> G:=CyclicGroup(8);
> <pc group of size 8 with 3 generators>
> gap> Display(CharacterTable(G));
> CT1
>
>      2  3   3  3  3   3   3  3   3
>
>        1a  8a 4a 2a  8b  8c 4b  8d
>
> X.1     1   1  1  1   1   1  1   1
> X.2     1  -1  1  1  -1  -1  1  -1
> X.3     1   A -1  1  -A   A -1  -A
> X.4     1  -A -1  1   A  -A -1   A
> X.5     1   B  A -1 -/B  -B -A  /B
> X.6     1  -B  A -1  /B   B -A -/B
> X.7     1 -/B -A -1   B  /B  A  -B
> X.8     1  /B -A -1  -B -/B  A   B
>
> A = E(4)
>   = Sqrt(-1) = i
> B = E(8)

See the following results in my example

gap> f := FreeGroup( "p", "q");;
gap> g42:= f/[ [ f.1 , f.1^-1 ], [ f.2 , f.2^-1 ], [ f.2 *f.1, f.1 *f.2 ] ];
<fp group on the generators [ p, q ]>


gap> Display(CharacterTable(g42));
CT1

     2  2  2  2  2

       1a 2a 2b 2c
    2P 1a 1a 1a 1a

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

Now, I'm confused on the following lines shown above:

     2  2  2  2  2

       1a 2a 2b 2c
    2P 1a 1a 1a 1a

1. What's the meaning of all 2's in this line?

     2  2  2  2  2

2. What's the meaning of 1a, 2a, 2b, 2c, and 2P, respectively?

> Cheers,
> Bill.