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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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