Ringel duality for perverse sheaves on hypertoric varieties

TL;DR

This paper establishes a duality in categories of perverse sheaves on hypertoric varieties, connecting algebraic and geometric structures, and confirms a conjecture in the context of symplectic singularities.

Contribution

It proves that the category of perverse sheaves on hypertoric varieties is a highest weight category with a Ringel dual, confirming a conjecture and linking to combinatorial algebra.

Findings

The category is a highest weight category.

Ringel duality relates hypertoric varieties and their Gale duals.

Confirmed the Finkelberg-Kubrak conjecture for hypertoric varieties.

Abstract

Motivated by the polynomial representation theory of the general linear group and the theory of symplectic singularities, we study a category of perverse sheaves with coefficients in a field $k$ on any affine unimodular hypertoric variety. Our main result is that this is a highest weight category whose Ringel dual is the corresponding category for the Gale dual hypertoric variety. On the way to proving our main result, we confirm a conjecture of Finkelberg-Kubrak in the case of hypertoric varieties. We also show that our category is equivalent to representations of a combinatorially-defined algebra, recently introduced in a related paper.