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Dear GAP Forum,

Allan Adler asked some technical and some theoretical questions

about character tables of alternating and symmetric groups.

For the theoretical ones, the book by James and Kerber on the

representation theory of symmetric groups should be a good reference.

For the technical questions, I try to give some answers.

GAP has the ATLAS character tables. Presumably, if asked to

generate a character table for the symmetric group on n

letters, without actually telling GAP that this was the group,

it would compute the same character table as in the ATLAS,

including the ordering of the conjugacy classes and of the

characters.Is that correct?

No.

Anyway, my question is about some orderings on the conjugacy classes

and characters of the symmetric group Sn. The ordering of the conjugacy

classes of Sn is basically an ordering on the partitions of n.

Do the orderings provided by the ATLAS and by GAP (in case they

are not the same) have a simple description?

In the ATLAS, the classes of symmetric groups are ordered such that

the classes in the alternating group precede the ones outside the

alternating group.

The classes in each of the two portions are ordered according to

increasing element order and for classes of same element order

according to decreasing centralizer order.

GAP provides several ways to generate character tables of symmetric

groups.

1. The tables stored in the table library have the same ordering of

classes and characters as the tables in the ATLAS (see the manual

section `ATLAS Tables' for some conventions concerning fusions and

extensions);

such a table is returned by `CharTable( "A5.2" )', the tables of

symmetric groups of degree up to 13 are available this way.

2. The tables created from the generic character table of symmetric

groups have the ordering of classes and characters given by partitions

as stored in the components `classparam' and `irredinfo' of the table

record;

such a table is returned by `CharTable( "Symmetric", 5 )',

the degree of the symmetric group is not bounded in this case.

3. If one computes the table from a group <G> by `CharTable( <G> )'

then it is only guaranteed that the class of the identity is the

first one.

Similarly, an ordering of the characters corresponds to an ordering

of the partitions of n. Is there a simple description of the ordering

of partitions corresponding to the character tables produced by

GAP or the ATLAS for Sn?

In ATLAS tables of simple groups, the characters are ordered according

to increasing degree.

For tables of automorphism groups of simple groups, the characters are

sorted according to the ordering of the constituents of their restrictions

to the simple group, again see the section `ATLAS Tables' of the manual.

For the ordering of characters in GAP tables, see the above statements

about the ordering of classes.

Kind regards

Thomas

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