Symmetric products of dg categories and semi-orthogonal decompositions

Naoki Koseki

Published 2022-05-19Version 1

In this article, we investigate semi-orthogonal decompositions of the symmetric products of dg-enhanced triangulated categories. Given a semi-orthogonal decomposition $\mathcal{D}=\langle \mathcal{A}, \mathcal{B} \rangle$, we construct semi-orthogonal decompositions of the symmetric products of $\mathcal{D}$ in terms of that of $\mathcal{A}$ and $\mathcal{B}$. This was originally stated by Galkin--Shinder, and answers the question raised by Ganter--Kapranov. We give two applications of our main result. Firstly, combining the above result with the derived McKay correspondence, we obtain various interesting semi-orthogonal decompositions of the derived categories of the Hilbert schemes of points on surfaces. Secondly, we prove the compatibility of our semi-orthogonal decompositions with categorical Heisenberg actions on the symmetric products of dg-categories due to Gyenge--Koppensteiner--Logvinenko. Using this compatibility, we prove the blow-up formula for the Heisenberg representations on the Grothendieck groups of the Hilbert schemes of points.