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@DrorSpeiser: the irreducible characters of $S_n$ of degree $n-1$ (for $n>6$) are $\chi$ and $sign \chi$, where $\chi$ is defined as above. The restriction of $\chi$ to $A_n$ is irreducible, so it remains to prove its uniqueness.
Thanks for your response. Result 2 of Rasala's paper is only saying that $n-1$ is the second least irreducible degree character of $S_n$ (not $A_n$). It can be shown that $n-1$ can be occured as the degree of an irreducible character of $A_n$. My question is about the uniqueness of such a character of degree $n-1$ for all $n>7$ for $A_n$. I know that the character is $\chi$ (see above, my comment for Dror Speiser). I would like to know if a implicit reference for the latter. Thanks to ban of Elsevier by some brave people, the paper can be freely downloaded from the publihser.
@Dror: The character $\chi$ is defined by $\chi(g)=|fix(g)|-1$ for all $g\in A_n$, where $fix(g)$ is the set of fixed points of $g$ on its natural action on $\lbrace 1,\dots,n\rbrace$. The restriction of sign times $\chi$ is the same as $\chi$ on $A_n$.
