> < ^ Date: Fri, 20 Jul 2001 17:39:22 -0400 (EDT)
> < ^ From: Vahid Dabbaghianabdoly <vdabbagh@math.carleton.ca >
> < ^ Subject: Re: Character Value

Dear GAP Froum;

```In the following group and character table , how can I find the value of
X.3 on the element
g:=[ [ Z(2^3)^6, Z(2^3)^6 ], [ Z(2^3)^6, Z(2^3)^5 ] ] in G.

gap> G:=SL(2,8);
SL(2,8)
gap> Display(CharacterTable(last));
SL(2,8)
```
```2  3  3  .  .  .  .  .  .  .
3  2  .  2  2  2  2  .  .  .
7  1  .  .  .  .  .  1  1  1
```
```1a 2a 3a 9a 9b 9c 7a 7b 7c
```
```X.1     1  1  1  1  1  1  1  1  1
X.2     7 -1 -2  1  1  1  .  .  .
X.3     7 -1  1  A  C  B  .  .  .
X.4     7 -1  1  B  A  C  .  .  .
X.5     7 -1  1  C  B  A  .  .  .
X.6     8  . -1 -1 -1 -1  1  1  1
X.7     9  1  .  .  .  .  D  E  F
X.8     9  1  .  .  .  .  E  F  D
X.9     9  1  .  .  .  .  F  D  E

A = E(9)^2+E(9)^4+E(9)^5+E(9)^7
B = -E(9)^4-E(9)^5
C = -E(9)^2-E(9)^7
D = E(7)^2+E(7)^5
E = E(7)^3+E(7)^4
F = E(7)+E(7)^6
```

Regards
Vahid Dabbaghian

On Fri, 20 Jul 2001, Alexander Hulpke wrote:

Dear GAP Forum,

How can I get the character value of an element of group G,
without checking that this element is in which conjugacy class
of G.

This depends a bit ob the representation G is given in, as well as the
representation that affords the character:

If G is a matrix group, and the character is given by the natural matrix
representation of G, you can use TraceMat.
If the character is given by the natural permutation representation, you can
check the number of fixed points:

Number(MovedPoints(G),x->x^g=x);

Otherwise you'll somehow have to connect the group and the representation.
If you can define the representation as a homomorphism, you can use the trace of
the image of an element (though you should be aware that for very large groups
the calculation of homomorphism images might be tedious).

Otherwise I don't see a way how you could (even in theory) get the character
value without (at least implicitly) identifying the class of the element.

what group and what character you are looking at.

Best wishes,

Alexander Hulpke

-- Colorado State University, Department of Mathematics,
Weber Building, Fort Collins, CO 80523, USA
email: hulpke@math.colostate.edu