> < ^ Date: Mon, 22 Feb 1993 07:28:27 +0100
> < ^ From: Jacob Hirbawi <hirbawi@commquest.com >
^ Subject: Direct character calculation

gap-forum@samson.math.rwth-aachen.de
Just a couple of comments in regards to my question last week about the
direct calculation of characters. First: many thanks to Thomas Breuer
for a great answer; now I am tempted to bother the forum with more
questions :-) . Second I have just finished installing gap3.2 on a
DEC Ultrix workstation and on a PC. It looks fantastic! it took just a
few minutes on the PC to do the calculations below for the characters
of <2,3,4> and <2,3,5> -- it took the DEC 3100 a bit longer to do the same.

```a := AbstractGenerator("a");
b := AbstractGenerator("b");
c := AbstractGenerator("c");
z := AbstractGenerator("z");
g4 := Group(a,b,c,z);
g5 := Group(a,b,c,z);
g4.relators := [a^2*z^-1,b^3*z^-1,c^4*z^-1,a*b*c*z^-1,z^2];
g5.relators := [a^2*z^-1,b^3*z^-1,c^5*z^-1,a*b*c*z^-1,z^2];
group4:=OperationCosetsFpGroup(g4,Subgroup(g4,[IdWord]));
group5:=OperationCosetsFpGroup(g5,Subgroup(g5,[IdWord]));
chr4 := CharTable(group4);
chr5 := CharTable(group5);
```

DisplayCharTable(chr4);

```2  4  2  1   3  4   3  1  3
3  1  .  1   .  1   .  1  .
```
```   1a 4a 6a  8a 2a  8b 3a 4b
2P 1a 2a 3a  4b 1a  4b 3a 2a
3P 1a 4a 2a  8b 2a  8a 1a 4b
5P 1a 4a 6a  8b 2a  8a 3a 4b
7P 1a 4a 6a  8a 2a  8b 3a 4b
```
```X.1   1  1  1   1  1   1  1  1
X.2   1 -1  1  -1  1  -1  1  1
X.3   2  . -1   .  2   . -1  2
X.4   2  .  1   A -2  -A -1  .
X.5   2  .  1  -A -2   A -1  .
X.6   3 -1  .   1  3   1  . -1
X.7   3  1  .  -1  3  -1  . -1
X.8   4  . -1   . -4   .  1  .

A = -E(8)+E(8)^3
= -ER(2) = -r2
DisplayCharTable(chr5);
```
```2  3  2  1   1  3   1  1   1   1
3  1  .  1   .  1   .  1   .   .
5  1  .  .   1  1   1  .   1   1
```
```   1a 4a 6a 10a 2a  5a 3a  5b 10b
2P 1a 2a 3a  5b 1a  5b 3a  5a  5a
3P 1a 4a 2a 10b 2a  5b 1a  5a 10a
5P 1a 4a 6a  2a 2a  1a 3a  1a  2a
7P 1a 4a 6a 10b 2a  5b 3a  5a 10a
```
```X.1   1  1  1   1  1   1  1   1   1
X.2   2  .  1   A -2  -A -1 -*A  *A
X.3   2  .  1  *A -2 -*A -1  -A   A
X.4   3 -1  .   A  3   A  .  *A  *A
X.5   3 -1  .  *A  3  *A  .   A   A
X.6   4  .  1  -1  4  -1  1  -1  -1
X.7   4  . -1   1 -4  -1  1  -1   1
X.8   5  1 -1   .  5   . -1   .   .
X.9   6  .  .  -1 -6   1  .   1  -1

A = -E(5)^2-E(5)^3
= (1+ER(5))/2 = 1+b5
```

Thanks to all the developers for a great tool.

Jacob Hirbawi
JcbHrb@CERF.net

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