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