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In his message of 8 May 93 Kay Magaard described some difficulties

with the computation of permutation characters for the symmetric group

$S_6$ and asks:

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);

S42 3 2 3 . 2 3 1 . . 1 .1a 2a 2b 3a 4a 2P 1a 1a 1a 3a 2b 3P 1a 2a 2b 1a 4aX.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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