Calabi-Yau properties of Postnikov diagrams

Abstract

We show that the dimer algebra of a connected Postnikov diagram in the disc is bimodule internally 3-Calabi-Yau in the sense of the author's earlier work [31]. As a consequence, we obtain an additive categorification of the cluster algebra associated to the diagram, which (after inverting frozen variables) is isomorphic to the homogeneous coordinate ring of a positroid variety in the Grassmannian, by a recent result of Galashin and Lam [13].

Special Tilting Modules for Algebras With Positive Dominant Dimension

“…To do this, we will use categorifications of the combinatorial ingredients introduced in Sections 2 and 3, and in this section we recall the necessary details of these constructions. For a connected Postnikov diagram D, the cluster algebra A D was categorified by the author [44], with the link to the cluster structures on C[ Π • P ] clarified in work with Çanakçı and King [7]. Perfect matchings and Muller-Speyer's left twist automorphism of Π • P are also categorified in [7].…”

“…Remark 5.14. Because A D is Noetherian (being free and finitely generated over the Noetherian ring Z) and of finite global dimension [44,Thm. 3.7], it follows from Theorems 5.12 and 5.13 together with a result of Kalck, Iyama, Wemyss and Yang [23, Thm.…”

“…Perhaps surprisingly, our proof relies primarily on representation theory and homological algebra. For connected positroids, the two cluster algebra structures have been categorified by the author [44], and the main algebraic fact underpinning the proof is that the two categorifications are derived equivalent in an extremely natural way. This gives us access to a homological method for detecting quasi-equivalences of cluster algebras, developed recently by Fraser and Keller [27,Appendix].…”

From frieze patterns to cluster categories

“…In this way we aim to illustrate how each successively more abstract framework-first passing from friezes to cluster algebras, and then to cluster categories-can lead to clean explanations of phenomena appearing at the previous level. This path can be followed further than we have space to do here, and leads to, for example, the theory of quivers with potential [24] and their more general cluster categories [2], Frobenius cluster categories and their applications to cluster algebras appearing in geometry [12,41,52,64,65], Adachi-Iyama-Reiten's τ -tilting theory [1], and the theory of stability conditions and scattering diagrams [10,11,45].…”

An introduction to relative Calabi-Yau structures

“…-applications in the study of Fukaya categories (cf. for example Brav-Dyckerhoff's [4,3]), -the categorification of cluster algebras with coefficients (as in the work of Geiss-Leclerc-Schröer [6], Leclerc [19], Jensen-King-Su [12], Pressland [22,21,20,23] . .…”