[GAP Forum] How to find the value of a character over some element in a group

azhvan sanna azhvan at hotmail.com
Fri Jul 10 21:18:46 BST 2009


Dear GAP Forum,

Thanks for you answers, I can say I got disappointed for sporadic groups other than Mathieu groups, as there is no way I can find out about uniquely which columns correspond to the value of my specific character on the given elements.
Basically I have to evaluate something like " Sigma_{g1, g2,..,gt} Xi(g1.g2.g3...gt) " for each Xi in Irr(G) and t from 1 to the degree of Xi.
So I need to find a way to calculate the "Xi(g1.g2.g3...gt)" quickly.
Is there any idea?

Cheers,
Azhvan

> Date: Wed, 8 Jul 2009 10:20:02 +0200
> From: thomas.breuer at math.rwth-aachen.de
> To: forum at gap-system.org
> Subject: Re: [GAP Forum] How to find the value of character over generators
> CC: 
> 
> Dear GAP Forum,
> 
> azhvan sanna wrote
> 
> > I have to use the character table of sporadic simple groups, and evaluate the
> > value of characters on some specific generators, I appreciate if somebody tell
> > me; is this possible? I have got the idea that as we do not know about the
> > Conjugacy classes ordering and the nature of them used for making characters
> > table, so characters table can be used when we do some calculation regardless
> > of element involve in evaluating, so computing for example the value of one
> > character in some element seems not possible.
> 
> 1. If one has computed a character table from a group in GAP
>    then the bijection between the columns of the table
>    and the conjugacy classes of this group is explicitly stored.
>    In this situation, a given character <chi> can be evaluated at a
>    group element <g> using <g>^<chi>, as in the following example.
> 
>         gap> G:= AlternatingGroup( 5 );;
>         gap> g:= (1,2,3,4,5);;
>         gap> tbl:= CharacterTable( G );;
>         gap> Display( tbl );
>         CT1
> 
>              2  2  2  .  .  .
>              3  1  .  1  .  .
>              5  1  .  .  1  1
> 
>                1a 2a 3a 5a 5b
>             2P 1a 1a 3a 5b 5a
>             3P 1a 2a 1a 5b 5a
>             5P 1a 2a 3a 1a 1a
> 
>         X.1     1  1  1  1  1
>         X.2     3 -1  .  A *A
>         X.3     3 -1  . *A  A
>         X.4     4  .  1 -1 -1
>         X.5     5  1 -1  .  .
> 
>         A = -E(5)-E(5)^4
>           = (1-ER(5))/2 = -b5
>         gap> chi:= Irr( tbl )[2];
>         Character( CharacterTable( Alt( [ 1 .. 5 ] ) ),
>         [ 3, -1, 0, -E(5)-E(5)^4, -E(5)^2-E(5)^3 ] )
>         gap> g^chi;
>         -E(5)-E(5)^4
> 
>    Note that in this situation GAP has to find out in which conjugacy class
>    of the group the given element lies.
>    So if one wants to evaluate several characters of a given character table
>    at the same group element, it is more efficient to compute once in which
>    conjugacy class this element lies, and then to fetch the character value
>    at the position of this class.
> 
>         gap> pos:= PositionProperty( ConjugacyClasses( tbl ), C -> g in C );
>         4
>         gap> chi[ pos ];
>         -E(5)-E(5)^4
> 
> 2. The situation is different if the bijection of table columns and
>    conjugacy classes is not explicitly known.
>    The question was about sporadic simple groups, the character tables
>    of these groups are printed in the famous Atlas of Finite Groups,
>    and these Atlas tables are available in GAP's Library of Character Tables.
>    Let us assume we are interested in the Mathieu group M_{11}.
>    Its Atlas character table can be fetched as follows.
> 
>        gap> tbl:= CharacterTable( "M11" );
>        CharacterTable( "M11" )
>        gap> Display( tbl );
>        M11
> 
>              2  4  4  1  3  .  1  3  3   .   .
>              3  2  1  2  .  .  1  .  .   .   .
>              5  1  .  .  .  1  .  .  .   .   .
>             11  1  .  .  .  .  .  .  .   1   1
> 
>                1a 2a 3a 4a 5a 6a 8a 8b 11a 11b
>             2P 1a 1a 3a 2a 5a 3a 4a 4a 11b 11a
>             3P 1a 2a 1a 4a 5a 2a 8a 8b 11a 11b
>             5P 1a 2a 3a 4a 1a 6a 8b 8a 11a 11b
>            11P 1a 2a 3a 4a 5a 6a 8a 8b  1a  1a
> 
>        X.1      1  1  1  1  1  1  1  1   1   1
>        X.2     10  2  1  2  . -1  .  .  -1  -1
>        X.3     10 -2  1  .  .  1  A -A  -1  -1
>        X.4     10 -2  1  .  .  1 -A  A  -1  -1
>        X.5     11  3  2 -1  1  . -1 -1   .   .
>        X.6     16  . -2  .  1  .  .  .   B  /B
>        X.7     16  . -2  .  1  .  .  .  /B   B
>        X.8     44  4 -1  . -1  1  .  .   .   .
>        X.9     45 -3  .  1  .  . -1 -1   1   1
>        X.10    55 -1  1 -1  . -1  1  1   .   .
> 
>        A = E(8)+E(8)^3
>          = ER(-2) = i2
>        B = E(11)+E(11)^3+E(11)^4+E(11)^5+E(11)^9
>          = (-1+ER(-11))/2 = b11
> 
>    If we are now given a representation of the group M_{11} and want to
>    use the Atlas character table together with this concrete group
>    then the bijection between the conjugacy classes of the group and the
>    columns of the Atlas table is not known.
>    For the first six columns of the table, the element orders determine
>    the corresponding conjugacy class uniquely,
>    but for an element of order 8 or 11, it is not clear to which column
>    of the table its conjugacy class corresponds.
> 
>    In this example, there is in fact no problem because of symmetries
>    of the character table, which allow one to choose any column of the
>    appropriate element order as the one that corresponds to a given element.
>    (But after that choice, the correspondence is of course fixed.)
> 
>    In general, it may happen that the correspondence between the classes
>    of a given group and the columns of the character table of this group
>    cannot be determined uniquely (up to symmetries of the table) without
>    additional information.
>    In the worst case, one may be forced to recompute (parts of) the character
>    table from the group.
>    However, often this is not necessary, and one does in fact not need the
>    complete information about the correspondence between classes and columns.
> 
> 3. The question was about sporadic simple groups,
>    and for most of these groups, the Atlas of Group Representations
>    (see http://brauer.maths.qmul.ac.uk/Atlas/)
>    provides a program for computing standard conjugacy class representatives,
>    if one starts with standard generators of the group in question.
>    In GAP, one can access this information via the package AtlasRep,
>    as follows.
>    We choose the degree 11 permutation representation of the Mathieu group
>    M_{11}.
> 
>        gap> LoadPackage( "atlasrep" );
>        true
>        gap> prg:= AtlasStraightLineProgram( "M11", "classes" );;
>        gap> info:= OneAtlasGeneratingSetInfo( "M11" );;
>        gap> gens:= AtlasGenerators( info );;
>        gap> reps:= ResultOfStraightLineProgram( prg.program, gens.generators );
>        [ (), (2,10)(4,11)(5,7)(8,9), (2,6,10)(4,8,7)(5,9,11),
>          (1,4,3,8)(2,5,6,9), (2,5,3,7,10)(4,6,11,8,9),
>          (1,3)(2,10,6)(4,11,8,5,7,9), (1,8,6,11,3,7,10,9)(4,5),
>          (1,7,6,9,3,8,10,11)(4,5), (1,4,11,3,8,2,10,5,7,6,9),
>          (1,8,6,5,7,2,10,9,3,4,11) ]
> 
>    The class representatives obtained this way correspond to the columns of
>    the Atlas character table of the group M_{11}.
> 
> 4. More information about character tables in GAP can be found in the
>    GAP Reference Manual, see for example
>       http://www.gap-system.org/Manuals/doc/htm/ref/CHAP069.htm
>    or try `?Character Tables' in GAP.
> 
> All the best,
> Thomas
> 
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