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

TL;DR

This paper constructs a natural basis for the algebra of regular functions on the $ ext{SL}_3$-character variety of a surface, using tropical coordinates linked to Fock-Goncharov theory and combinatorial graph data.

Contribution

It introduces a new tropical coordinate system for the basis elements of the $ ext{SL}_3$-character variety, connecting combinatorial graph invariants with geometric tropical points.

Findings

Basis elements indexed by non-negative integer coordinates

Coordinates satisfy Knutson-Tao rhombus inequalities and modulo 3 conditions

Coordinates relate to tropical points at infinity of the dual character variety

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.