Categorification of ice quiver mutation

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.Comment: 34 pages; arXiv admin note: text overlap with arXiv:0906.0761, arXiv:1810.01179 by other authors; many improvements and corrections;to appear in Mathematische Zeitschrif