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Newton-Okounkov bodies obtained from certain orbits of plabic graphs

Michael Schlößer

Abstract

We investigate the plabic graphs corresponding to the quadrilateral Postnikov arrangements used by J.Scott to equip the homogeneous coordinate rings of Grassmannians with a cluster structure. More precisely we describe their orbits under the natural action of the dihedral group and show that the associated Newton-Okounkov bodies are all unimodular equivalent to Gelfand-Tsetlin polytopes.

Newton-Okounkov bodies obtained from certain orbits of plabic graphs

Abstract

We investigate the plabic graphs corresponding to the quadrilateral Postnikov arrangements used by J.Scott to equip the homogeneous coordinate rings of Grassmannians with a cluster structure. More precisely we describe their orbits under the natural action of the dihedral group and show that the associated Newton-Okounkov bodies are all unimodular equivalent to Gelfand-Tsetlin polytopes.
Paper Structure (10 sections, 33 theorems, 62 equations, 9 figures)

This paper contains 10 sections, 33 theorems, 62 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Plabic graphs
  3. Rectangles and Checkboards
  4. Dihedral symmetry
  5. Quivers
  6. Grassmannians
  7. $\mathcal{A}$-cluster structure
  8. The Superpotential and Newton-Okounkov bodies
  9. The superpotential and (dual) checkboards
  10. Newton-Okounkov bodies of (dual) checkboards

Key Result

Proposition 1.1

Let $\rho \in S_n$, then $\rho$ preserves weak separation if and only if $\rho \in D_n$. In particular $D_n$ acts on the set of maximal weakly separated collections.

Figures (9)

  • Figure 1: A plabic graph with labelled faces.
  • Figure 2: Rules of the road: turn left at white vertices and right at black vertices
  • Figure 3: The rectangle graph $G^\text{rec}_{5,9}$ with its face labels $I_{i,j}$ arranged in a grid.
  • Figure 4: The checkboard graph $G^\text{ch}_{5,9}$ with its face labels $I_{i,j}$ arranged in a grid.
  • Figure 5: The edges contributing to the label of the face $f_{2,3}$ in row $2$ and column $3$ of the checkboard graph $G^\text{ch}_{6,10}$.
  • ...and 4 more figures

Theorems & Definitions (67)

  • Proposition 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Conjecture 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.6
  • Definition 2.7
  • ...and 57 more