Grassmannian cluster subcategories和positroid varieties
Grassmannian cluster subcategories and positroid varieties
AI总结：
本文研究Grassmannian cluster subcategories与positroid varieties的关系，证明GP B具有cluster substructure并构造了cluster algebra A_clu，同时展示了其与坐标环的联系。
摘要：
本文构造了Grassmannian cluster子类GP B，并证明其具有CM C_{k,n}的cluster子结构，从而得到cluster algebra A_clu。同时展示了A_clu与坐标环以及B对应的positroid variety的坐标环之间的联系。利用cluster子结构，我们证明了cluster tilting对象U∈GP B的ice Gabriel quiver Q°_U是Q°_T的子quiver，其中T是包含U作为summand的CM C_{k,n}中的cluster tilting对象。我们还推导出A_clu是C[Gr(k,n)]的子代数。此外，应用Canakci-King-Pressland关于Gabriel quiver Q_U在B连通情况下的结果，我们推导出Q°_U对于任意B都与Muller-Speyer从plabic图构造的quiver一致。因此，局部化代数(A_clu)_B与Muller-Speyer的cluster algebra A_MS同构。我们还构造了某些子代数和C[Gr(k,n)]的理想的基础，并应用这些来证明(A_clu)_B自然同构于开positroid variety的坐标环。作为结果，我们获得了Galashin-Lam定理的新证明，该定理将A_MS与开positroid variety的坐标环识别，最初由Muller-Speyer猜想。在连通情况下，我们还注意到Pressland给出了Galashin-Lam之后的cluster结构的categoryification。
英文摘要：
A class of subcategories GP $B$ of the Grassmannian cluster category CM $C_{k, n}$ was constructed by Jensen--King--Su from certain superorders $B$ of $C_{k, n}$, which they showed are in bijection with Grassmannian positroids of type $(k, n)$. We prove that GP $B$ admits a cluster substructure of CM $C_{k, n}$, giving rise to a cluster algebra $A_{clu}$. This naturally raises questions regarding the relationship of $A_{clu}$ to $C[Gr(k, n)]$ and to the coordinate ring of the positroid variety associated to $B$. Using the cluster substructure, we show that the ice Gabriel quiver $Q^\circ_U$ of a cluster tilting object $U\in$ GP $B$, consisting of rank one modules, is a subquiver of $Q^\circ_T$ with $T$ a cluster tilting object in CM $C_{k, n}$ containing $U$ as a summand. We also deduce that $A_{clu}$ is a subalgebra of $C[Gr(k, n)]$. Moreover, applying a result of Canakci--King--Pressland on the Gabriel quiver $Q_U$ in the case where $B$ is connected (i.e., has no repeated direct summands), we deduce that $Q^\circ_U$, for arbitrary $B$, coincides with the quiver constructed by Muller-Speyer from a plabic graph whose face labels agree with the indices of the indecomposable summands of $U$. Consequently, the localised algebra $(A_{clu})_B$ is isomorphic to the cluster algebra $A_{MS}$ of Muller-Speyer. We then construct bases for certain subalgebras and for an ideal of $C[Gr(k, n)]$, and apply these to prove that $(A_{clu})_B$ is naturally isomorphic to the coordinate ring of the open positroid variety. As a consequence, we obtain a new proof of Galashin--Lam's Theorem, identifying $A_{MS}$ with the coordinate ring of the open positroid variety, which was originally conjectured by Muller-Speyer. In the connected case, we note also that Pressland gave a categorification of the cluster structure following Galashin-Lam.
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