Cell Decompositions for Rank Two Quiver Grassmannians

TL;DR

This paper proves that all quiver Grassmannians for exceptional representations of a generalized Kronecker quiver have a cell decomposition, linking combinatorics, geometry, and cluster algebra theory.

Contribution

It introduces a new class of regular representations called truncated preprojectives and establishes their cell decompositions, connecting combinatorial labelings with geometric structures.

Findings

Cell decompositions for quiver Grassmannians of exceptional representations.

Bijection between combinatorial labelings and geometric cells.

Connection between Dyck path combinatorics and 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.