Combinatorial constructions of derived equivalences

TL;DR

This paper develops explicit combinatorial methods to construct derived equivalences between GIT quotients of quasi-symmetric representations, revealing new wall crossing phenomena and actions of the K"ahler moduli space on derived categories.

Contribution

It introduces an algorithmic approach to describe tilting bundles, basis of K-theory, and fundamental groupoid actions for derived categories of GIT quotients in the context of quasi-symmetric representations.

Findings

Constructed explicit tilting bundles generating derived categories.

Provided combinatorial bases for K-theory of GIT quotients.

Established actions of the K"ahler moduli space on derived categories.

Abstract

Given a certain kind of linear representation of a reductive group, referred to as a quasi-symmetric representation in recent work of \v{S}penko and Van den Bergh, we construct equivalences between the derived categories of coherent sheaves of its various geometric invariant theory (GIT) quotients for suitably generic stability parameters. These variations of GIT quotient are examples of more complicated wall crossings than the balanced wall crossings studied in recent work on derived categories and variation of GIT quotients. Our construction is algorithmic and quite explicit, allowing us to: 1) describe a tilting vector bundle which generates the derived category of such a GIT quotient, 2) provide a combinatorial basis for the K-theory of the GIT quotient in terms of the representation theory of G, and 3) show that our derived equivalences satisfy certain relations, leading to a representation of the fundamental groupoid of a "K\"ahler moduli space" on the derived category of such a GIT quotient. Finally, we use graded categories of singularities to construct derived equivalences between all Deligne-Mumford hyperk\"ahler quotients of a symplectic linear representation of a reductive group (at the zero fiber of the algebraic moment map and subject to a certain genericity hypothesis on the representation), and we likewise construct actions of the fundamental groupoid of the corresponding K\"ahler moduli space.