Playlist - Dimer models and cluster categories of Grassmannians

The homogeneous coordinate ring of the Grassmannian Gr(k,n) has a beautiful structure as a cluster algebra, by a result of J. Scott. Central to this description is a collection of clusters containing only Plücker coordinates, which are described by certain diagrams in a disk, known as Postnikov diagrams or alternating strand diagrams. Recent work of B. Jensen, A. King and X. Su has shown that the Frobenius category of Cohen-Macaulay modules over a certain algebra, B, can be used to categorify this structure.

In joint work with Karin Baur and Alastair King, we associate a dimer algebra A(D) to a Postnikov diagram D, by interpreting D as a dimer model with boundary. We show that A(D) is isomorphic to the endomorphism algebra of a corresponding Cohen-Macaulay cluster-tilting B-module, i.e. that it is a cluster-tilted algebra in this context. The proof uses the consistency of the dimer model in an essential way.

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 disk. The general surface case can also be considered, and we compute boundary algebras associated to the annulus.

The homogeneous coordinate ring of the Grassmannian Gr(k,n) has a beautiful structure as a cluster algebra, by a result of J. Scott. Central to this description is a collection of clusters containing only Plücker coordinates, which are described by certain diagrams in a disk, known as Postnikov diagrams or alternating strand diagrams. Recent work of B. Jensen, A. King and X. Su has shown that the Frobenius category of Cohen-Macaulay modules over a certain algebra, B, can be used to categorify this structure.

In joint work with Karin Baur and Alastair King, we associate a dimer algebra A(D) to a Postnikov diagram D, by interpreting D as a dimer model with boundary. We show that A(D) is isomorphic to the endomorphism algebra of a corresponding Cohen-Macaulay cluster-tilting B-module, i.e. that it is a cluster-tilted algebra in this context. The proof uses the consistency of the dimer model in an essential way.

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 disk. The general surface case can also be considered, and we compute boundary algebras associated to the annulus.

The homogeneous coordinate ring of the Grassmannian Gr(k,n) has a beautiful structure as a cluster algebra, by a result of J. Scott. Central to this description is a collection of clusters containing only Plücker coordinates, which are described by certain diagrams in a disk, known as Postnikov diagrams or alternating strand diagrams. Recent work of B. Jensen, A. King and X. Su has shown that the Frobenius category of Cohen-Macaulay modules over a certain algebra, B, can be used to categorify this structure.

In joint work with Karin Baur and Alastair King, we associate a dimer algebra A(D) to a Postnikov diagram D, by interpreting D as a dimer model with boundary. We show that A(D) is isomorphic to the endomorphism algebra of a corresponding Cohen-Macaulay cluster-tilting B-module, i.e. that it is a cluster-tilted algebra in this context. The proof uses the consistency of the dimer model in an essential way.

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 disk. The general surface case can also be considered, and we compute boundary algebras associated to the annulus.