[GAP Forum] character tables ordering

[dmitrii.pasechnik at cs.ox.ac.uk]

Dear all,
On Thu, Feb 02, 2017 at 01:18:12PM +0100, Frank Lübeck wrote:
> 
> I don't see how one could define some "standard" or "canonical" ordering
> on conjugacy classes or irreducible characters of any finite group.

Assuming one agrees on a standard ordering of the cyclotomic numbers
involved (e.g. the one used by GAP),
one can then order the tables so that the result is
lexicographiically maximal (or minimal), row-wise (or column-wise).
Pick your preferred flavour of the latter alternatives.

Best,
Dima

> 
> 
> On Thu, Feb 02, 2017 at 11:35:47AM +0100, Bill Allombert wrote:
> > Dear GAP Forum,
> > 
> > If I do:
> > gap> S4:=SymmetricGroup(4);
> > Sym( [ 1 .. 4 ] )
> > gap> Display(CharacterTable(S4));
> > CT1
> > 
> >      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
> 
> In this case GAP "knows" that S4 is a symmetric group on 4 letters and
> returns the library table for this group. It contains the standard
> labelling of classes and characters by partitions of 4:
> 
> t1 := CharacterTable(S4);
> ClassParameters(t1);
> CharacterParameters(t1);
> InfoText(t1);
> 
> The ordering for rows and colums is that of Partitions(4).
> 
> > but if I do:
> > gap> IdGroup(S4);
> > [ 24, 12 ]
> > gap> G:=SmallGroup(24,12);
> > <pc group of size 24 with 4 generators>
> > gap> Display(CharacterTable(G));
> > CT2
> > 
> >      2  3  2  .  3  2
> >      3  1  .  1  .  .
> > 
> >        1a 2a 3a 2b 4a
> >     2P 1a 1a 3a 1a 2b
> >     3P 1a 2a 1a 2b 4a
> > 
> > X.1     1  1  1  1  1
> > X.2     1 -1  1  1 -1
> > X.3     2  . -1  2  .
> > X.4     3 -1  . -1  1
> > X.5     3  1  . -1 -1
> > 
> > so I get two different ordering for the row and columns of the table,
> > which is not entirely surprising since different algorithms are used
> > here.
> 
> If GAP does not know about an appropriate library table and computes the
> table with some (here Dixon's) algorithm then the tables can even be different 
> on different occasions. Also the labelling of conjugacy classes may be different.
> 
> t2 := CharacterTable(G);
> InfoText(t2);
> 
> > My question is whether there are some standard order for printing
> > character table. This would be useful for interoperability between
> > software (or two version of the same software) by using a common
> > numbering for characters which could avoid mistakes.
> 
> To check if two tables are the same modulo permutation of rows and columns
> you can use 
> 
> TransformingPermutationsCharacterTables(t1,t2);
> 
> For the display of character tables you can use some optional hints
> collected in a record, see
> 
> ?Reference: Display (for a character table)
> ?Browse: Browse (for character tables)
> 
> For example, to display characters with respect to '<' of rows:
> 
> ord := [1..Length(Irr(t1))];;
> SortParallel(ShallowCopy(Irr(t1)),ord);
> Display(t1, rec(chars := ord));
> Browse(t1, rec(chars := ord));
> 
> With best regards,
>    Frank
> -- 
> ///  Dr. Frank Lübeck, Lehrstuhl D für Mathematik, Pontdriesch 14/16,
> \\\                    52062 Aachen, Germany
> ///  E-mail: Frank.Luebeck at Math.RWTH-Aachen.De
> \\\  WWW:    http://www.math.rwth-aachen.de/~Frank.Luebeck/
> 
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