Dear Alexander and John, dear Forum,
thanks a lot for the helpful answers. I misinterpreted the function
"TransformingPermutationsCharacterTables", as I thought it just performs
the permutation on the rows and columns of the character table, but it
also takes care of the power maps. Hence, this is exactly the function
I was looking for with my first question.
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)?
(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 <abac...@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
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