Is it true that in each row and column of the character table of alternating groups with degree $\geq 7$ there are at most two complex values? Any reference will be highly appreciated.



Character Values for Alternating Groups of degree $\geq 7$
As Geoff thought, the answer is contained in James and Kerber (it's Theorem 2.5.13 in "The Representation Theory of the Symmetric Group", Encyclopedia of Mathematics and its Applications vol. 16, 1981).


Each row or column of the character table contains at most one pair of irrationalities.


If $\lambda$ is a self-conjugate partition (and so corresponds to an irreducible character of $S\_n$ that splits on restriction to $A\_n$), then the only cycle type of elements where the corresponding character values are irrational is the one with cycle lengths equal to the lengths of the hooks centred on the main diagonal of the Young diagram (notice that these are all odd and distinct, so correspond to a conjugacy class of $S\_n$ that splits in $A\_n$).


For example, for the partition $(6,6,6,6,6,6)$ this cycle type would be $[11,9,7,5,3,1]$.


