Quiver Grassmannians of extended Dynkin type D - Part 2: Schubert decompositions and F-polynomials
Quiver Grassmannians of extended Dynkin type D - Part 2: Schubert decompositions and F-polynomials
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扩展 Dynkin D 型的 Quiver Grassmannians - 第 2 部分：舒伯特分解和 F 多项式
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
Thorsten Weist
中科院分区：
文献类型：
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作者：
Oliver Lorscheid;Thorsten Weist
推广了第一部分的主要结果，在本文的第一部分中，我们证明了扩展的dykin型箭图的表示的每个箭图Grassman都有到仿射空间的分解。在小缺陷的实根表示的情况下，非空单元与固定树形系数的顶点集的某些子集一一对应，即所谓的非矛盾子集。在第二部分中，我们利用这个刻画来确定箭袋Grassmannians的欧拉特征的母函数。$F$-多项式)。沿着这些思路，我们得到了来自扩展的dykin型箭图的簇代数的所有簇变量的显式公式。
Extending the main result of Part 1, in the first part of this paper we show that every quiver Grassmannian of a representation of a quiver of extended Dynkin type $D$ has a decomposition into affine spaces. In the case of real root representations of small defect, the non-empty cells are in one-to-one correspondence to certain, so called non-contradictory, subsets of the vertex set of a fixed tree-shaped coefficient quiver. In the second part, we use this characterization to determine the generating functions of the Euler characteristics of the quiver Grassmannians (resp. $F$-polynomials). Along these lines, we obtain explicit formulae for all cluster variables of cluster algebras coming from quivers of extended Dynkin type $D$.