> < ^ Date: Wed, 27 Aug 1997 18:09:22 -0400
> ^ Subject: ordering of conjugacy classes of Sn

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?

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?

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?

Using the original definition of Frobenius, I can in principle
compute a character table of Sn, with the conjugacy classes
and characters arranged according to a natural ordering of the
partitions, e.g. decreasing lexicographic ordering. I wrote
a program to do this but it is very slow, since it involves
multiplying a lot of polyomials and looking at coefficients
(i.e. the original construction of Frobenius). It got as far
as S5 and I shut it off after a few hours of trying to compute
S6. With S5, the characters are grouped into 3 sections. The
middle section happens to have only one character and it is
of maximum degree. The other two sections have their characters
arranged symmetrically around the middle, the symmetry being
tensor with the sign character. There is a particular
odd permutation in this case, namely a 4 cycle, such that the first
group consists of all characters which are positive on that
element.

Is there a similar description in general for Frobenius' character table?
Is there always a conjugacy class that defines the sections as above?

Is there a general prescription for passing from the character table
that GAP or ATLAS would produce and the one described above?

In the case of S5, the class of the 4 cycle is in the middle
of the list of three classes of odd permutations. For S6, S7 the
number of classes of odd permutations is also odd. I don't know
if that is true in general. If so, the middle one is again a
tempting choice. For S6 the middle one is again a 4 cycle
and for S7 it is a 6 cycle.

True or false: for the ordering of GAP or ATLAS, the middle
conjugacy class of odd permutations is represented by an
n-2 cycle if n is even and by an n-1 cycle if n is odd?

The middle element makes a tempting choice for a "defining element",
but there are some characters which vanish on it, at least for S6 and S7,
and those characters need not be decomposable when restricted
to the alternating groups.

I would appreciate it if someone more knowledgeable about these
very naive questions can shed some light on them.