Cyclic Characters of Alternating Groups

We determine the eigenvalues with multiplicity of each element of an alternating group in any irreducible representation. This is equivalent to determining the decomposition of cyclic representations of alternating groups into irreducibles. We characterize pairs $(w, V)$, where $w$ is an element and $V$ is an irreducible representation of an alternating group such that $w$ admits a non-zero invariant vector in $V$. We also establish large new families of global conjugacy classes for alternating groups, thereby giving a new proof of a result of Heide and Zalessky on the existence of such classes.