Quasi-coincidence of cluster structures on positroid varieties

TL;DR

The paper resolves a fundamental question about the relationship between two natural cluster structures on open positroid varieties—source-labelled and target-labelled—by proving they quasi-coincide. The authors deploy a categorification framework using two Calabi–Yau Frobenius categories derived from the dimer/boundary algebras associated to Postnikov diagrams, and then apply Fraser–Keller theory to obtain a quasi-cluster equivalence between the two structures. They further show that Muller–Speyer’s left twist is itself a quasi-cluster morphism between these structures. The reduction to connected positroids and the correspondence between perfect matchings, Plücker coordinates, and cluster variables provide a robust, category-theoretic pathway to relate seed data across different labellings, deepening the link between positroid geometry and cluster algebras. Overall, the work unifies geometric and categorical perspectives and clarifies how twists and labellings influence cluster structures on positroid varieties.

Abstract

By work of a number of authors, beginning with Scott and culminating with Galashin and Lam, the coordinate rings of positroid varieties in the Grassmannian carry cluster algebra structures. In fact, they typically carry many such structures, the two best understood being the source-labelled and target-labelled structures, referring to how the initial cluster is computed from a Postnikov diagram or plabic graph. In this article, we show that these two cluster algebra structures quasi-coincide, meaning in particular that a cluster variable in one structure may be expressed in the other structure as the product of a cluster variable and a Laurent monomial in the frozen variables. This resolves a conjecture attributed to Muller and Speyer from 2017. The proof depends critically on categorification: of the relevant cluster algebra structures by the author, of perfect matchings and twists by the author with Çanakçı and King, and of quasi-equivalences of cluster algebras by Fraser and Keller. By similar techniques, we also show that Muller and Speyer's left twist map is a quasi-cluster equivalence from the target-labelled structure to the source-labelled structure.

Figures (10)

• Figure 1: A Postnikov diagram and its corresponding plabic graph.
• Figure 2: A plabic graph of type $(3,7)$. The left-hand figure shows the source labels, while the right-hand figure shows the target labels. The labels of boundary regions, which form the necklaces, are displayed in blue since these will later label the frozen Plücker coordinates in a cluster structure (see Theorem \ref{['t:GalLam']}).
• Figure 3: The downstream (magenta) and upstream (blue) wedge of an edge in a plabic graph $\Gamma$. While \ref{['d:P4']} guarantees that these regions are indeed wedge-shaped, the right-hand figure demonstrates that they may not be disjoint.
• Figure 4: The arrow of $Q_D$ at a crossing of $D$, or equivalently at an edge of $\Gamma_D$. The rules at half-edges are obtained by cutting the figure in half along the arrow.
• Figure 5: A plabic graph $D$ for a positroid variety in $\mathrm{Gr}_{3}^{7}$ with its dual quiver. The quiver vertices are given their source labels on the left, and their target labels on the right (cf. Figure \ref{['f:labels']}).

Theorems & Definitions (141)

• Definition 2.1: Postnikov-PosGrass
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Remark 2.5
• Definition 2.6
• Remark 2.7
• Proposition 2.8
• proof
• Corollary 2.9