> < ^ Date: Mon, 10 May 1993 15:34:22 +0200
> < ^ From: Goetz Pfeiffer <Goetz.Pfeiffer@NUIGalway.ie >
> < ^ Subject: Re: PermChars

In his message of 8 May 93 Kay Magaard described some difficulties
with the computation of permutation characters for the symmetric group

Does a generic character table need to be reordered before
asking for permutaion characters?

Sometimes, yes. 'PermChars' expects the 1-character to be the first
one in the list 'irreducibles'. This is not the case for tables
constructed by the generic table of symmetric groups. These tables
are constructed in such a way that the labelling of the characters
coincides with the labelling of the conjugacy classes. And the
partition \$[1^n]\$ which labels the identity is the label for the sign
character, not for the trivial one. (I will use the smaller table of
the group \$S_4\$ as an example.)

gap> s4:= CharTable("Symmetric", 4);;
gap> DisplayCharTable(s4);
S4

```2  3  2  3  .  2
3  1  .  .  1  .
```
```   1a 2a 2b 3a 4a
2P 1a 1a 1a 3a 2b
3P 1a 2a 2b 1a 4a
```
```X.1   1 -1  1  1 -1
X.2   3 -1 -1  .  1
X.3   2  .  2 -1  .
X.4   3  1 -1  . -1
X.5   1  1  1  1  1

gap> s4.classparam;
[ [ 1, [ 1, 1, 1, 1 ] ], [ 1, [ 2, 1, 1 ] ], [ 1, [ 2, 2 ] ],
[ 1, [ 3, 1 ] ], [ 1, [ 4 ] ] ]
gap> s4.irredinfo;
[ rec(
charparam := [ 1, [ 1, 1, 1, 1 ] ] ), rec(
charparam := [ 1, [ 2, 1, 1 ] ] ), rec(
charparam := [ 1, [ 2, 2 ] ] ), rec(
charparam := [ 1, [ 3, 1 ] ] ), rec(
charparam := [ 1, [ 4 ] ] ) ]
```

After exchanging the first and the last character in the table 's4' by
the function 'SortCharactersCharTable' the function 'PermChars' will
do its job.

```gap> SortCharactersCharTable(s4, (1,5));
(1,5)
gap> PermChars(s4, 4);
[ [ 4, 0, 4, 1, 0 ], [ 4, 2, 0, 1, 0 ] ]
```

There will be a more precise error message, if the 1-character is not
in the expected position, in the next version.

Goetz Pfeiffer.

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