Lefschetz exceptional collections in $$S_k$$-equivariant categories of $$(\mathbb {P}^n)^{k}$$

Abstract

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 of maximal length 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 full 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 a full minimal Lefschetz exceptional collection for \(n=1\) and even k, and for \(n=2\) and \(k=3\).