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Introduction to Cluster Algebras. Chapter 7

Sergey Fomin, Lauren Williams, Andrei Zelevinsky

TL;DR

This chapter develops plabic graphs as a combinatorial framework encoding cluster-algebraic data, unifying their local moves with quiver mutations and highlighting their connections to triangulations and wiring diagrams. It establishes a robust correspondence between reduced plabic graphs, triple diagrams, and decorated/affine permutations, via the fundamental theorem and the normal/plabic-triple bijection. It then develops two complementary labeling schemes—edge-resonance and face-labels—and shows how bridge decompositions and affine permutations parameterize reduced graphs, enabling a precise classification through Grassmann necklaces and weakly separated collections. Together, these results provide a unified toolkit for studying cluster structures in Grassmannians and related varieties, with broad ties to triangulations, wiring diagrams, and positroid combinatorics.

Abstract

This is a preliminary draft of Chapter 7 of our forthcoming textbook "Introduction to Cluster Algebras." Chapters 1-3 have been posted as arXiv:1608.05735. Chapters 4-5 have been posted as arXiv:1707.07190. Chapter 6 has been posted as arXiv:2008.09189. This installment contains: Chapter 7. Plabic graphs

Introduction to Cluster Algebras. Chapter 7

TL;DR

This chapter develops plabic graphs as a combinatorial framework encoding cluster-algebraic data, unifying their local moves with quiver mutations and highlighting their connections to triangulations and wiring diagrams. It establishes a robust correspondence between reduced plabic graphs, triple diagrams, and decorated/affine permutations, via the fundamental theorem and the normal/plabic-triple bijection. It then develops two complementary labeling schemes—edge-resonance and face-labels—and shows how bridge decompositions and affine permutations parameterize reduced graphs, enabling a precise classification through Grassmann necklaces and weakly separated collections. Together, these results provide a unified toolkit for studying cluster structures in Grassmannians and related varieties, with broad ties to triangulations, wiring diagrams, and positroid combinatorics.

Abstract

This is a preliminary draft of Chapter 7 of our forthcoming textbook "Introduction to Cluster Algebras." Chapters 1-3 have been posted as arXiv:1608.05735. Chapters 4-5 have been posted as arXiv:1707.07190. Chapter 6 has been posted as arXiv:2008.09189. This installment contains: Chapter 7. Plabic graphs

Paper Structure

This paper contains 14 sections, 51 theorems, 22 equations, 47 figures.

Table of Contents

  1. Plabic graphs
  2. Plabic graphs and the main results
  3. Plabic graphs and their quivers
  4. Triangulations and wiring diagrams via plabic graphs
  5. Trivalent plabic graphs
  6. Triple diagrams and normal plabic graphs
  7. Minimal triple diagrams
  8. From minimal triple diagrams to reduced plabic graphs
  9. The bad features criterion
  10. Affine permutations
  11. Bridge decompositions
  12. Edge labels of reduced plabic graphs
  13. Face labels of reduced plabic graphs
  14. Grassmann necklaces and weakly separated collections

Key Result

Proposition 7.1.17

Let $G$ be a reduced leafless plabic graph. If $\pi_G(i) = i$, then the connected component of $G$ containing the boundary vertex $i$ is a lollipop.

Figures (47)

  • Figure 7.1: (a) A plabic graph $G$. (b) A plabic graph $G'$ with a white lollipop.
  • Figure 7.5: A leafless plabic graph is not reduced if and only if it is move-equivalent to a graph containing a hollow monogon or digon.
  • Figure 7.6: Two reduced plabic graphs sharing the same decorated trip permutation $(3,4,5,1,2,\overline 6)$. Cf. \ref{['fig:plabic']}(b).
  • Figure 7.7: This plabic graph $G$ with one boundary vertex has a single face but is not reduced. Note that it is not leafless.
  • Figure 7.8: Two plabic graphs and their associated quivers. Shown on the left is the graph $G$ from \ref{['fig:plabic']}(a). The quiver on the right has double arrows, corresponding to the instances where a pair of faces share two boundary segments disconnected from each other. The frozen vertex $v$ at the top of the picture is isolated: the two arrows between $v$ and an internal vertex located underneath $v$ cancel each other.
  • ...and 42 more figures

Theorems & Definitions (145)

  • Definition 7.1.1
  • Remark 7.1.2
  • Remark 7.1.3
  • Definition 7.1.4
  • Definition 7.1.5
  • Definition 7.1.6
  • Remark 7.1.7
  • Definition 7.1.8
  • Remark 7.1.9
  • Remark 7.1.10
  • ...and 135 more