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Tropical Fock-Goncharov coordinates for $\mathrm{SL}_3$-webs on surfaces I: construction

Daniel C. Douglas, Zhe Sun

Abstract

For a finite-type surface $\mathfrak{S}$, we study a preferred basis for the commutative algebra $\mathbb{C}[\mathscr{R}_{\mathrm{SL}_3(\mathbb{C})}(\mathfrak{S})]$ of regular functions on the $\mathrm{SL}_3(\mathbb{C})$-character variety, introduced by Sikora-Westbury. These basis elements come from the trace functions associated to certain tri-valent graphs embedded in the surface $\mathfrak{S}$. We show that this basis can be naturally indexed by non-negative integer coordinates, defined by Knutson-Tao rhombus inequalities and modulo 3 congruence conditions. These coordinates are related, by the geometric theory of Fock and Goncharov, to the tropical points at infinity of the dual version of the character variety.

Tropical Fock-Goncharov coordinates for $\mathrm{SL}_3$-webs on surfaces I: construction

Abstract

For a finite-type surface , we study a preferred basis for the commutative algebra of regular functions on the -character variety, introduced by Sikora-Westbury. These basis elements come from the trace functions associated to certain tri-valent graphs embedded in the surface . We show that this basis can be naturally indexed by non-negative integer coordinates, defined by Knutson-Tao rhombus inequalities and modulo 3 congruence conditions. These coordinates are related, by the geometric theory of Fock and Goncharov, to the tropical points at infinity of the dual version of the character variety.

Paper Structure

This paper contains 57 sections, 32 theorems, 58 equations, 58 figures.

Table of Contents

  1. Introduction
  2. Global webs
  3. Topological setting
  4. Webs
  5. Faces
  6. Non-elliptic webs
  7. Local webs
  8. Ideal polygons
  9. Local webs
  10. External faces
  11. Combinatorial identity
  12. Non-ellipticity
  13. Essential and rung-less local webs
  14. Ladder-webs in ideal biangles
  15. Honeycomb-webs in ideal triangles
  16. ...and 42 more sections

Key Result

Theorem 1

For a punctured finite-type surface $\mathfrak{S}$ equipped with an ideal triangulation $\lambda$, the Sikora-Westbury $\mathrm{SL}_3$-web basis for the algebra of functions $\mathbb{C}[\mathscr{R}_{\mathrm{SL}_3(\mathbb{C})}(\mathfrak{S})]$ admits an explicit system of non-negative integer coordina

Figures (58)

  • Figure 1: Ideal triangulations
  • Figure 2: Web
  • Figure 3: Global parallel-move
  • Figure 4: Prohibited square-face
  • Figure 5: Local webs
  • ...and 53 more figures

Theorems & Definitions (115)

  • Theorem 1
  • Theorem 2: DouglasArxiv20b
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Definition 9
  • Proposition 10: compare KuperbergCommMathPhys96
  • ...and 105 more