[GAP Forum] Comparing character tables in GAP

[Alexander Hulpke]

Dear Forum,

> For the second question I had the following situation in mind:
> 
> gap> c1 := CharacterTable(AlternatingGroup(6));
> CharacterTable( Alt( [ 1 .. 6 ] ) )
> gap> c2 := CharacterTable("A6") mod 2;
> BrauerTable( "A6", 2 )
> 
> So c1 and c2 are two character tables of the alternating group of degree 6.  Is there a GAP-function checking if the two tables (ordinary character tables or Brauer tables) are tables of isomorphic groups?  Or do I have to be lucky that for both tables the attribute UnderlyingGroup(-) is known (respectively UnderlyingGroup(OrdinaryTable(-)), if it is a Brauer table)?

If you get character tables (by name) from the character table library, tables are guaranteed to be consistent, same names mean the same group and class namings are consistent between ordinary and Brauer tables.

But if you calculate one of the tables from a group your question actually would require a classification of all groups with these tables to ensure that they would have to be isomorphic (and could not be Brauer Pairs), so there is no such algorithm. (Of course in many cases theory will tell you that the groups are isomorphic, but this is impossible to put into an algorithm.)

So in your example, if you set

c1:=CharacterTable("A6");

you are fine, but as given there is no guarantee.

Best,

    Alexander Hulpke






> (I am aware of the possibility to call "CharacterTableWithStoredGroup" to attach a group to a character table, but I was wondering if this is not somehow done automatically for tables coming from the ATLAS?)
> 
> Thanks again for your answers!
> 
> Kind regards,
>  Andreas
> 
> 
> On 2014-07-01 17:36, Alexander Hulpke wrote:
>> Dear Forum,
>> On Jul 1, 2014, at 7/1/14 1:44, Andreas Bächle <ABachle at vub.ac.be> wrote:
>>> I want to check whether two character tables are "equal" (in the sense that their irreducible characters coincide and also the corresponding power maps match, of course, everything up to permutation).  According to Thomas Breuer's answer from 2000 (http://www.gap-system.org/ForumArchive/Breuer.1/Thomas.1/Re__Comp.8/1.html) there is no built in function to compare character tables in GAP,
>> I think this is a misunderstanding. The archived email says that the
>> only way to test such equivalence is by an explicit search for
>> permutations, there is no cheaper test that would only need to check
>> some properties without possibly testing for permutations.
>>> but one has to do the checks manually.  Is this still the case?  What would be the best way to do it?  In particular, I want to recognize, for example, the two character tables CharacterTable("A4") and CharacterTable(AlternatingGroup(4)) as "equal".  As I want to use this in a function, I want to automatize it as far as possible.
>> gap> c1:=CharacterTable("A4");
>> CharacterTable( "a4" )
>> gap> c2:=CharacterTable(AlternatingGroup(4));
>> CharacterTable( Alt( [ 1 .. 4 ] ) )
>> gap> TransformingPermutationsCharacterTables(c1,c2);
>> rec( columns := (), group := Group([ (3,4) ]), rows := (2,3) )
>> The result tells you that the classes are already in correspondence,
>> but characters 2 and 3 were swapped.
>>> In case only the "heads" of the character tables are known and I want to avoid to calculate the irreducibles (as the groups might be very large and the calculations needed can be done with induced characters), there is probably no hope to see if two character tables are tables of the same group, except if the attribute "UnderlyingGroup" is stored for both tables?
>> Since there are Brauer Pairs you will never be able to determine
>> group isomorphism based on equivalence of the character tables (or do
>> I misunderstand the question)?
>> Best,
>>   Alexander Hulpke