Dimer models and cluster categories of Grassmannians (Q2827968)

In order to study the total positivity of the Grassmannian \(Gr(k,n)\) of \(k\)-planes in \(\mathbb{C}^n\), \textit{A. Postnikov} [``Total positivity, Grassmannians, and networks'', Preprint, \url{arXiv:0609764}] introduced Postnikov diagrams (or alternating strand diagrams) as collections of curves in a disk satisfying certain axioms, which was used by \textit{J. S. Scott} [Proc. Lond. Math. Soc. (3) 92, No. 2, 345--380 (2006; Zbl 1088.22009)] to show the cluster algebra structure of the homogeneous coordinate ring of \(\mathrm{Gr}(k,n)\).NEWLINENEWLINE\textit{C. Geiß} et al. [Ann. Inst. Fourier 58, No. 3, 825--876 (2008; Zbl 1151.16009)] obtained an additive categorification of this cluster algebra structure in terms of a subcategory of the category of finite dimensional modules over the preprojective algebra of type \(A_{n-1}\). However, a single cluster coefficient, the minor corresponding to the k-subset \(\{1,2, \dots, k\}\) is not realised in the category. Recently \textit{B. T. Jensen} et al. [Proc. Lond. Math. Soc. (3) 113, No. 2, 185--212 (2016; Zbl 1375.13033)] provided a full and direct categorification of the cluster structure on the homogeneous coordinate ring, using the category of (maximal) Cohen-Macaulay modules over the completion of an algebra \(B\), which is a quotient of the preprojective algebra of type \(\tilde{A}_{n-1}\). In particular, a rank one Cohen-Macaulay \(B\)-module \(\mathbb{M}_I\) is associated to every \(k\)-subset \(I\) of \(\{1,2, \ldots ,n\}\).NEWLINENEWLINEGiven a Postnikov diagram \(D\), let \(T_D = \bigoplus \mathbb{M}_I\), where the direct sum is over the \(I\) labelling the alternating regions of \(D\). Note that \(T_D\) is a certain Cohen-Macaulay module over the algebra \(B\). For the Postnikov diagram \(D\), the authors first construct a quiver with faces \(Q(D)\), which is a dimer model. Then the authors associated the dimer algebra \(A_D=A_{Q(D)}\) to \(Q(D)\). The main result of the paper is that \(\text{End}_{B}(T_D)\) is isomorphic to the dimer algebra \(A_D\).