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An explicit description for quiver Grassmannians of the Kronecker quiver

Ulysses Alvarez, Kyungyong Lee

Abstract

It is an open problem to find cell decompositions of quiver Grassmannians associated to each cluster variable. We initiate a new approach to this problem by giving an explicit description for each individual subrepresentation. In this paper, we illustrate our approach for the Kronecker quiver.

An explicit description for quiver Grassmannians of the Kronecker quiver

Abstract

It is an open problem to find cell decompositions of quiver Grassmannians associated to each cluster variable. We initiate a new approach to this problem by giving an explicit description for each individual subrepresentation. In this paper, we illustrate our approach for the Kronecker quiver.
Paper Structure (9 sections, 6 theorems, 79 equations)

This paper contains 9 sections, 6 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Cell decompositions of quiver Grassmannians
  3. Quiver Grassmannians
  4. The matrices $N_2(P)$ and $N_1(P)$
  5. The Fibonacci trees
  6. The main theorem
  7. Proofs
  8. Proof of $\mathcal{K}(P)\subseteq\mathcal{J}(P)$
  9. Proof of $\mathcal{J}(P)\subseteq\mathcal{K}(P)$

Key Result

Theorem 1.1

The coefficients of any cluster variable in the Laurent expansion are positive integers.

Theorems & Definitions (33)

  • Theorem 1.1: LS4GHKKDav
  • Theorem 1.2: DWZ2
  • Theorem 2.1
  • Definition 3.1
  • Example 3.2
  • Example 3.3
  • Definition 3.4
  • Remark 3.5
  • Example 3.6
  • Definition 4.1
  • ...and 23 more