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Cell decompositions for rank two quiver Grassmannians. (English) Zbl 07238508
Let $$K(n)$$ denote the quiver with $$2$$ vertices and $$n$$ arrows, all in the same direction. This paper studies quiver Grassmannians of $$K(n)$$, the algebraic varieties parametrising $$\mathbf{e}$$-dimensional subrepresentations of $$V$$, for a given $$K(n)$$-representation $$V$$ and given dimension vector $$\mathbf{e}$$, and shows that for certain $$V$$ this variety admits a cell decomposition. In particular, this is shown whenever $$V$$ is a direct sum of exceptional representations (and so in particular for the indecomposable preprojective or preinjective representations) or for $$V$$ a truncated preprojective representation, meaning the cokernel of a suitable map of preprojective representations.
The proof strategy is to show that these Grassmannians are smooth, and then to consider their fixed points under an appropriate $$\mathbb{C}^*$$-action. This fixed points turn out to be quiver Grassmannians for representations of the universal abelian covering quiver of $$K(n)$$. Iterating this process, one eventually obtains quiver Grassmannians for which a cell decomposition can be constructed, implying the existence of such a decomposition for the original Grassmannian. The implementation of this strategy relies on a careful and detailed study of the representation theory of the very special family of quivers $$K(n)$$.
The authors also describe two combinatorial ways of indexing the cells in these decompositions, the first using successor closed subsets of a $$2$$-quiver, and the second using compatible pairs in a maximal Dyck path. Both of these results allow one to compute the Euler characteristic of the quiver Grassmannian as the cardinality of a set, thus giving an explanation of its positivity, which is important in the theory of cluster algebras, cf. [K. Lee and R. Schiffler, Ann. of Math. (2) 182, No. 1, 73–125 (2015; Zbl 1350.13024)]

MSC:
 16G20 Representations of quivers and partially ordered sets 14M15 Grassmannians, Schubert varieties, flag manifolds 13F60 Cluster algebras
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References:
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