TITLE: Character Values for Alternating Groups of degree $\geq 7$
QUESTION [7 upvotes]: 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.

REPLY [2 votes]: This answer essentially summarizes information from the other answers, hopefully, making the whole picture clear. For each self-conjugate partition $\lambda$ (i.e., $\lambda=\lambda'$) of $n$, the irreducible character $\chi_\lambda$ of $S_n$ is a sum of two irreducible representations of $A_n$: $\chi_\lambda = \chi_\lambda^+ + \chi_\lambda^-$. Also, for each partition with distinct odd parts $\mu=(2m_1+1,\dotsc,2m_d+1)$, where $m_1>\dotsb >m_d$ the conjugacy class $C_\mu$ of elements with cycle type $\mu$ in $S_n$ splits into two $A_n$-conjugacy classes, represented by elements $w_\mu^+$ and $w_\mu^-$.
There is a bijective correspondence $\lambda\mapsto \phi(\lambda)$ from the set of self-conjugate partitions onto the set of partitions with distinct odd parts. The image of $\lambda$ is $\mu=(2m_1+1,\dotsc,2m_d+1)$ if the number of boxes in the Young diagram of $\lambda$ strictly to the right (or strictly below) a box of the form $(i,i)$ is $m_i$.
Theorem. All character values of $A_n$ are real except possibly those of the form $\chi_{\lambda}^\pm(w_\mu^\pm)$, where $\mu=\phi(\lambda)=(2m_1+1,\dotsc,2m_d+1)$.
The character value $\chi_{\lambda^\pm}(w_\mu^\pm)$ is not real if and only if $\sum_{i=1}^d m_i$ is odd.
It follows that, in each row and column of the character table there is at most one pair of irrationalities. The only rows with irrationalities are those corresponding to $\lambda^\pm$ where $\lambda$ is a self-conjugate partitions such that the number of off-diagonal boxes in its Young diagram is odd. The only columns with irrationalities are of the classes of elements with cycle decomposition $(2m_1+1,\dotsc,2m_d+1)$, where $m_1>\dotsb > m_d$, and $\sum_{i=1}^d m_d$ is odd.
The only alternating groups $A_n$ with real character table are for the values $n=2,5,6,10,14$.
James and Kerber discuss the characters of alternating groups, but do not give any interesting examples. Frobenius (Georg Ferdinand Frobenius. 1901. ‘Über die Charaktere der alternirenden Gruppe.’
[On the characters of alternating groups.] S’ber Akad. Wiss. Berlin, 303–15) has computations for $A_8$. My book has many examples and a more detailed explanation in Chapters 4 and 5.