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A gluing operation for dimer quivers

Karin Baur, Colin Krawchuk

Abstract

In this article we introduce a gluing operation on dimer models. This allows us to construct dimer quivers on arbitrary surfaces. We study how the associated dimer and boundary algebras behave under the gluing and how to determine them from the gluing components. We also use this operation to construct homogeneous dimer quivers on annuli.

A gluing operation for dimer quivers

Abstract

In this article we introduce a gluing operation on dimer models. This allows us to construct dimer quivers on arbitrary surfaces. We study how the associated dimer and boundary algebras behave under the gluing and how to determine them from the gluing components. We also use this operation to construct homogeneous dimer quivers on annuli.
Paper Structure (6 sections, 21 theorems, 45 equations, 13 figures)

This paper contains 6 sections, 21 theorems, 45 equations, 13 figures.

Table of Contents

  1. Dimer Quivers
  2. Postnikov Diagrams
  3. Diagrams on surfaces
  4. Dimer Gluing
  5. Bridge Quivers
  6. Gluing dimer quivers for annuli

Key Result

Lemma 1.8

Let $t$ be the element of $A_Q$ given by $t:= \sum_{i \in Q_0}u_i$, then $\mathbb{C}[t] \subseteq Z(A_Q)$, where $Z(A_Q)$ denotes the centre of $A_Q$.

Figures (13)

  • Figure 1: Associating a strand diagram to a dimer quiver
  • Figure 2: Paths $u$ (in blue) and $v$ (in red).
  • Figure 3: Two Postnikov diagrams on an annulus.
  • Figure 4: Dimer quivers of weak Postnikov diagrams.
  • Figure 5: The quivers $Q$ and $\rho(Q)$: adding boundary arrows.
  • ...and 8 more figures

Theorems & Definitions (86)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Example 1.4
  • Remark 1.5
  • Definition 1.6
  • Definition 1.7
  • Lemma 1.8
  • proof
  • Definition 1.9
  • ...and 76 more