Let $F$ be an algebraically closed field of characteristic $p \geq 0$ and $A_n$ be the alternating group on $n$ letters. The main goal of this paper is to describe the pairs $(G, E)$, where $E$ is an irreducible $FA_n$-module and $G < A_n$ is a proper subgroup of $A_n$ such that the restriction $E{\downarrow_G}$ is irreducible. We are able to give a list of all such pairs, provided $p > 3$. The case $p = 0$ has been treated by J. Saxl. The problem is important for the classification of maximal subgroups in finite classical groups.

2000 Mathematical Subject Classification: 20C20, 20C30, 20B35, 20B20.