Dimer models and cluster categories of Grassmannians
Dimer models and cluster categories of Grassmannians
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DOI：
10.1112/plms/pdw029
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发表时间：
2016-08-01
影响因子：
1.8
通讯作者：
Marsh, Bethany R.
中科院分区：
文献类型：
--
作者：
Baur, Karin;King, Alastair D.;Marsh, Bethany R.
我们将二聚体代数A与对应于Grassmannian Gr（k，n）的簇结构中的未成年人簇的Postnikov图D（在圆盘中）相关联。我们证明了A同构于代数B上对应的Cohen-Macaulay模T的自同态代数，B用于Jensen-King-Su对Gr（k，n）的簇结构进行分类.因此，B可以被实现为A的边界代数，即对应于圆盘边界的幂等元e的子代数eAe。构造和证明使用图D的解释，其相关的plabic图和对偶图（有面），作为一个二聚体模型与边界。我们还讨论了一般表面的情况下，特别是计算边界代数相关联的环。
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.