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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.

Allan Adler

ara@altdorf.ai.mit.edu

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