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Twists, Higher Dimer Covers, and Web Duality for Grassmannian Cluster Algebras

Esther Banaian, Elise Catania, Christian Gaetz, Miranda Moore, Gregg Musiker, Kayla Wright

Abstract

We study a twisted version of Fraser, Lam, and Le's higher boundary measurement map, using face weights instead of edge weights, thereby providing Laurent polynomial expansions, in Plücker coordinates, for twisted web immanants for Grassmannians. In some small cases, Fraser, Lam, and Le observe a phenomenon they call "web duality'', where web immanants coincide with web invariants, and they conjecture that this duality corresponds to transposing the standard Young tableaux that index basis webs. We show that this duality continues to hold for a large set of $\text{SL}_3$ and $\text{SL}_4$ webs. Combining this with our twisted higher boundary measurement map, we recover and extend formulas of Elkin-Musiker-Wright for twists of certain cluster variables. We also provide evidence supporting conjectures of Fomin-Pylyavskyy as well as one by Cheung-Dechant-He-Heyes-Hirst-Li concerning classification of cluster variables of low Plücker degree in $\mathbb{C}[\text{Gr}(3,n)]$.

Twists, Higher Dimer Covers, and Web Duality for Grassmannian Cluster Algebras

Abstract

We study a twisted version of Fraser, Lam, and Le's higher boundary measurement map, using face weights instead of edge weights, thereby providing Laurent polynomial expansions, in Plücker coordinates, for twisted web immanants for Grassmannians. In some small cases, Fraser, Lam, and Le observe a phenomenon they call "web duality'', where web immanants coincide with web invariants, and they conjecture that this duality corresponds to transposing the standard Young tableaux that index basis webs. We show that this duality continues to hold for a large set of and webs. Combining this with our twisted higher boundary measurement map, we recover and extend formulas of Elkin-Musiker-Wright for twists of certain cluster variables. We also provide evidence supporting conjectures of Fomin-Pylyavskyy as well as one by Cheung-Dechant-He-Heyes-Hirst-Li concerning classification of cluster variables of low Plücker degree in .

Paper Structure

This paper contains 10 sections, 13 theorems, 36 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Grassmannian cluster algebras
  4. The boundary measurement map
  5. Web invariants
  6. Skein relations on tensor diagrams
  7. Web bases and tableaux
  8. The Fraser--Lam--Le pairing and web duality
  9. Twists and $r$-dimer covers
  10. Computing web duals

Key Result

Proposition 2.11

For each standard $\mathop{\mathrm{SL}}\nolimits_3$ web $W$ of degree $(1^{12})$, we have

Figures (4)

  • Figure 1: A top cell plabic graph of type $(3,12)$.
  • Figure 2: A dual semistandard $\mathop{\mathrm{SL}}\nolimits_5$ web $W$ of degree $\lambda$, a semistandard $\mathop{\mathrm{SL}}\nolimits_5$ web $X$ of degree $\lambda$, and the unclasped web $\overline{X}$, where $\lambda = (3,2,1,2,1,2,1,1,1,1)\in\mathbb{N}^{10}$.
  • Figure 3: The unique consistent labelings $\mathcal{L}_W\in A(\mathcal{S}_W; W)$ and $\mathcal{L}_{\overline{X}}\in A(\mathcal{S}_{\overline{X}}; \overline{X})$ (see Example \ref{['ex:boundary-subsets']}). For each edge $\mathbf{e},$ the subset $L(\mathbf{e})\subset[5]$ is denoted by the subset of colors appearing on that edge.
  • Figure 4: The wrench relation for $\mathop{\mathrm{SL}}\nolimits_3$ tensor invariants in the FP sign convention.

Theorems & Definitions (40)

  • Definition 2.1: postnikov2006total
  • Definition 2.2: postnikov2006total
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Example 2.7
  • Definition 2.8: FLL19
  • Definition 2.9: FP16
  • Definition 2.10: Fomin--Pylyavskyy(FP) sign convention, FP16
  • ...and 30 more