A basic set for the alternating group

This article concerns the $p$-basic set existence problem in the representation theory of finite groups. We show that, for any odd prime $p$, the alternating group ${\text{A}}_n$ has a $p$-basic set. More precisely, we prove that the symmetric group ${\text{sym}}_n$ has a $p$-basic set with some additional properties, allowing us to deduce a $p$-basic set for ${\text{A}}_n$. Our main tool is the generalized perfect isometries introduced by K"ulshammer, Olsson and Robinson. As a consequence we obtain some results on the decomposition number of ${\text{A}}_n$.