# Order of symmetric group irreps

This is a key detail for me, but I can't find any information about it in the Sage or GAP documentation: if I use

SymmetricGroup(5).character_table()


I get a 7x7 matrix, with rows indexed by conjugacy classes and columns indexed by irreps. Both of these things can be indexed, for instance, by Young diagrams of size 5.

How are the rows and columns ordered? Is it safe to assume that they will be ordered in reverse-lexicographic order on the Young diagrams, i.e.

5 ; 4,1 ; 3,2 ; 3,1,1 ; 2,2,1 ; 2,1,1,1 ; 1,1,1,1,1

This seems to be the case in the examples I've checked but I want to be sure that this is guaranteed by the code and not a coincidence in small examples.

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( 2016-10-29 07:09:18 -0500 )edit

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According to the documentation on class functions,

NOTE: The ordering of the columns of the character table of a group corresponds to the ordering of the list. However, in general there is no way to canonically list (or index) the conjugacy classes of a group. Therefore the ordering of the columns of the character table of a group is somewhat random.

I've used this functionality a fair amount but to my surprise there doesn't seem to be good direct documentation of character_table easily discoverable online. You have to know that it lives here, and it doesn't say much more about it than to ask how GAP orders it.

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I figured as much, but the fact that it is not (and cannot be done) for general groups says nothing about whether it gives a consistent result for symmetric groups in particular! If there is no consistent order of the columns, then there must at least be some way of determining the order. If you can't, then it would be impossible to use the character table for almost anything!

(In particular, I am computing the character of a certain representation as a vector and want to use the character table to decompose it.)

( 2016-11-03 20:27:41 -0500 )edit

I've done the same type of computation as you, I think, and usually can figure it out - note the documentation says "corresponds to the ordering of the list". So I guess you can just have it list the conjugacy classes G.conjugacy_classes()? I'd try that.

( 2016-11-04 07:54:50 -0500 )edit