Categorification of ice quiver mutation

TL;DR

This paper extends the categorification of quiver mutation to ice quivers with potential, establishing derived equivalences of associated Ginzburg algebras and illustrating applications in dimer models and positroid cluster structures.

Contribution

It generalizes Keller and Yang's categorification to ice quivers with potential, linking it to relative Calabi-Yau structures and providing new examples and applications.

Findings

Derived equivalences between relative Ginzburg algebras for ice quivers with potential

Categorification of mutation at frozen vertices in positroid cluster structures

Applications to dimer models and Grassmannian cluster categories

Abstract

In 2009, Keller and Yang categorified quiver mutation by interpreting it in terms of equivalences between derived categories. Their approach was based on Ginzburg's Calabi-Yau algebras and on Derksen-Weyman-Zelevinsky's mutation of quivers with potential. Recently, Matthew Pressland has generalized mutation of quivers with potential to that of ice quivers with potential. In this paper, we show that his rule yields derived equivalences between the associated relative Ginzburg algebras, which are special cases of Yeung's deformed relative Calabi-Yau completions arising in the theory of relative Calabi-Yau structures due to To\"en and Brav-Dyckerhoff. We illustrate our results on examples arising in the work of Baur-King-Marsh on dimer models and cluster categories of Grassmannians. We also give a categorification of mutation at frozen vertices as it appears in recent work of Fraser-Sherman-Bennett on positroid cluster structures.