Dimer models and cluster categories of Grassmannians

Abstract

We associate a dimer algebra A to a Postnikov diagram D (in a disc) corresponding to a cluster of minors in the cluster structure of the Grassmannian Gr(k,n). We show that A is isomorphic to the endomorphism algebra of a corresponding Cohen–Macaulay module T over the algebra B used to categorify the cluster structure of Gr(k,n) by Jensen–King–Su. It follows that B can be realised as the boundary algebra of A, that is, the subalgebra eAe for an idempotent e corresponding to the boundary of the disc. The construction and proof uses an interpretation of the diagram D, with its associated plabic graph and dual quiver (with faces), as a dimer model with boundary. We also discuss the general surface case, in particular computing boundary algebras associated to the annulus.

Mutation of frozen Jacobian algebras

“…Originally appearing in the context of statistical mechanics [25,33], these constructions have been well-studied in the mathematics and physics literature; see for example [9,13,20,29] in the case that Σ is closed, and [3,16] in the general case.…”

“…More recently, it has been fruitful to enhance the data of a quiver with potential by declaring a subquiver to be frozen, leading to the more general notion of a frozen Jacobian algebra. These algebras appear naturally when considering dimer models on surfaces with boundary [16], as well as endomorphism algebras of cluster-tilting objects in Frobenius categorifications of cluster algebras with frozen variables [3,11,30]. The goal of the present paper is to survey some of the results on Jacobian algebras, particularly relating to their mutation, and at the same time fill a gap in the literature by extending these statements to the more general case of frozen Jacobian algebras.…”

Internally Calabi–Yau algebras and cluster-tilting objects

“…5 are inspired by work of Broomhead [7] on consistent dimer models (also known as brane tilings or bipartite field theories) on closed surfaces, which has applications to theoretical physics. We expect our results to have consequences for the more recent theory of dimer models on surfaces with boundary, studied for example by Franco [17], and which has already appeared in the context of cluster categorification for the Grassmannian in work of Baur-King-Marsh [5]. Dimer models on surfaces with boundary have also appeared, under the name 'plabic graphs', in work of Postnikov [37] on the positive Grassmannian, and in recent work of Goncharov [21].…”

“…Indeed, this is the case for at least some cluster-tilting objects in the families of Frobenius cluster categories we described in Examples 3.11 and 3.12; see [9,Thm. 6.6], [5,Thm. 10.3].…”

“…Baur-King-Marsh [5,Thm. 10.3] show that for certain cluster-tilting objects T ∈ CM(B), the endomorphism algebra End B (T ) op is isomorphic to a frozen Jacobian algebra (Definition 5.1) associated to a dimer model on a disk, with the projection onto a maximal projective-injective summand corresponding to the sum of idempotents at the frozen vertices.…”