arXiv 2402.145012024-02math.RTmath.CO

秩2网的聚类提升

Cluster Categorification of Rank 2 Webs

Mathematical Sciences Institute Australian National University（澳大利亚国立大学数学科学研究所）；International Center for Mathematical Sciences - Sofia, Bulgarian Academy of Sciences（保加利亚科学院索非亚国际数学中心）

Ian Le, Emine Yıldırım

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AI总结：

本文通过网的组合描述，推测性地给出了格拉斯曼流形Gr(k,n)中所有秩2聚类变量的提升。

摘要：

格拉斯曼流形Gr(k,n)的齐次坐标环具有众所周知的聚类结构。Jensen-King-Su等人通过一个环A_{k,n}的模类对这一聚类结构进行了提升，基于Geiss-Leclerc-Schröer的工作，在其中聚类变量对应于不可分解的刚性模。我们通过使用网来描述对应于秩2聚类变量的模，这推测性地给出了所有秩2聚类变量的提升。

英文摘要：

The homogeneous coordinate ring of the Grassmannian $\rm{Gr}(k,n)$ has a well-known cluster structure. There is a categorification of this cluster structure via a category of modules for a ring $A_{k,n}$ due to Jensen-King-Su, building on work of Geiss-Leclerc-Schröer, in which cluster variables correspond to indecomposable rigid modules. We give a combinatorial description of modules that correspond to rank $2$ cluster variables by using webs. This conjecturally gives the categorification of all rank $2$ cluster variables.

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