Perverse sheaves on symmetric products of the plane

TL;DR

This paper provides an algebraic description of perverse sheaves on symmetric products of the plane, relating them to modules over a new algebra akin to the Schur algebra, and extends Springer theory to Hilbert schemes.

Contribution

It introduces a novel algebraic framework for perverse sheaves on symmetric products of the plane, connecting them to a new algebra related to the Schur algebra and extending modular Springer theory.

Findings

Category of perverse sheaves is equivalent to modules over a new algebra

Established an analogue of modular Springer theory for Hilbert schemes

Provided an algebraic description valid over any field

Abstract

For any field $k$, we give an algebraic description of the category $\mathrm{Perv}_\mathscr{S}(S^n (\mathbb{C}^2),k)$ of perverse sheaves on the $n$-fold symmetric product of the plane $S^n(\mathbb{C}^2)$ constructible with respect to its natural stratification and with coefficients in $k$. In particular, we show that it is equivalent to the category of modules over a new algebra that is closely related to the Schur algebra. As part of our description we obtain an analogue of modular Springer theory for the Hilbert scheme $\mathrm{Hilb}^n(\mathbb{C}^2)$ of $n$ points in the plane with its Hilbert-Chow morphism.