# [GAP Forum] How to find the value of character over generators

Thomas Breuer thomas.breuer at math.rwth-aachen.de
Wed Jul 8 09:20:02 BST 2009

```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.
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
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.

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}.

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}.