  We determine the decomposition of cyclic characters of alternating groups
into irreducible characters. As an application, we characterize pairs $(w, V)$,
where $w\in A_n$ and $V$ is an irreducible representation of $A_n$ such that
$w$ admits a non-zero invariant vector in $V$. We also establish new global
conjugacy classes for alternating groups, thereby giving a new proof of a
result of Heide and Zalessky on the existence of such classes.
