  We consider the bounded derived category of $S_k$-equivariant coherent
sheaves on $(\mathbb{P}^n)^k$. The goal of this paper is to construct in this
category a rectangular Lefschetz exceptional collection when this is possible,
or a minimal Lefschetz exceptional collection when a rectangular one does not
exist. The main results of the paper include the construction of a rectangular
Lefschetz exceptional collection in the case $k=3$ and in the case $n=1$ when
$\mathrm{gcd}(n+1,k)=1$. We also construct minimal Lefschetz exceptional
collection for $n=1$ and even $k$, and for $n=2$ and $k=3$.
