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2016-11-03 20:27:41 -0600 commented answer Order of symmetric group irreps

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.)

2016-11-01 08:12:17 -0600 received badge  Student (source)
2016-10-29 07:07:56 -0600 asked a question 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.