Computation of irreducible characters for $S_n$ - Mathematica vs. GAP

Question

For some physical applications I need the knowledge of irreducible characters for symmetric groups $S_n$ with large $n$. For small ones I was using FiniteGroupData in Mathematica, but this is limited to $n=10$. For larger $n$ I am using GAP: Display(CharacterTable(SymmetricGroup(n))); (cf. Mathematica-stackexchange).

The result obtained by these two methods differ by signs. I was trying to find the origin of these differences in documentation of these programs but I didn't find the answer. Why the results from Mathematica (which agree with D. M. Jackson "Tables of Irreducible Characters of the Symmetric Group on at most Fifteen Symbols" 1988; see also here) differ from ones which are produced by using GAP? It is almost true that the difference is at even positions in the tables, but this pattern is not exact. Does anyone know how to convert GAP's result into the same convention as used in Mathematica?

Answer 1

Of course it is highly unlikely that the tables really differ, but the issue is most likely an arrangement of classes and characters.

CharacterTable(SymmetricGroup(n)) will take $S_n$ as a group and calculate the character table with generic methods (that might be somewhat adapted. There is no guarantee about the arrangement.

CharacterTable("S10") takes the table from the character table library. Here class/character arrangement is made to be compatible with the ATLAS of finite groups.

You probably want to use the parameterized constructor CharacterTable("Symmetric",10). This uses the Murnaghan-Nakayama relations to calculate the table. Most crucial (assuming the variable c holds the table), you can use ClassParameters(c) and CharacterParameters(c) to get the partitions corresponding to the classes, respectively characters. This will probably allow you to translate to the convention in other publications.

Answer 2

Are the two results really different? A character table is only determined up to ordering of the rows and columns. Did you check if the two results are the same up to such permutations ?