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Mathematics > Representation Theory

Title: Cell Decompositions for Rank Two Quiver Grassmannians

Abstract: We prove that all quiver Grassmannians for exceptional representations of a generalized Kronecker quiver admit a cell decomposition. In the process, we introduce a class of regular representations which arise as quotients of consecutive preprojective representations. Cell decompositions for quiver Grassmannians of these "truncated preprojectives" are also established. We also provide two natural combinatorial labelings for these cells. On the one hand, they are labeled by certain subsets of a so-called 2-quiver attached to a (truncated) preprojective representation. On the other hand, the cells are in bijection with compatible pairs in a maximal Dyck path as predicted by the theory of cluster algebras. The natural bijection between these two labelings gives a geometric explanation for the appearance of Dyck path combinatorics in the theory of quiver Grassmannians.
Comments: 35 pages
Subjects: Representation Theory (math.RT); Algebraic Geometry (math.AG); Rings and Algebras (math.RA)
MSC classes: 16G20, 14M15, 13F60
Cite as: arXiv:1803.06590 [math.RT]
  (or arXiv:1803.06590v1 [math.RT] for this version)

Submission history

From: Thorsten Weist [view email]
[v1] Sun, 18 Mar 2018 00:46:44 GMT (42kb)